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Page 1: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-DISTANCE-BALANCED GRAPHS

Bostjan FrelihUniversity of Primorska, Slovenia

Joint work with Stefko Miklavic

May 27, 2015

Page 2: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Motivation: Distance-balanced graphs

A graph Γ is said to be distance-balanced if for any edge uv of Γ,the number of vertices closer to u than to v is equal to the numberof vertices closer to v than to u.

A distance partition of a graph with diameter 4 with respect toedge uv .

Page 3: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Motivation: Distance-balanced graphs

I These graphs were first studied (at least implicitly) by K.Handa who considered distance-balanced partial cubes.

I The term itself is due to J. Jerebic, S. Klavzar and D. F. Rallwho studied distance-balanced graphs in the framework ofvarious kinds of graph products.

Page 4: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Motivation: Distance-balanced graphs

I These graphs were first studied (at least implicitly) by K.Handa who considered distance-balanced partial cubes.

I The term itself is due to J. Jerebic, S. Klavzar and D. F. Rallwho studied distance-balanced graphs in the framework ofvarious kinds of graph products.

Page 5: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

References

K. Handa, Bipartite graphs with balanced (a, b)-partitions, ArsCombin. 51 (1999), 113-119.

K. Kutnar, A. Malnic, D. Marusic, S. Miklavic,Distance-balanced graphs: symmetry conditions, DiscreteMath. 306 (2006), 1881-1894.

J. Jerebic, S. Klavzar, D. F. Rall, Distance-balanced graphs,Ann. comb. (Print. ed.) 12 (2008), 71-79.

Page 6: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

References

K. Kutnar, A. Malnic, D. Marusic, S. Miklavic, The stronglydistance-balanced property of the generalized Petersen graphs.Ars mathematica contemporanea. (Print. ed.) 2 (2009), 41-47.

A. Ilic, S. Klavzar, M. Milanovic, On distance-balanced graphs,Eur. j. comb. 31 (2010), 733-737.

S. Miklavic, P. Sparl, On the connectivity of bipartitedistance-balanced graphs, Europ. j. Combin. 33 (2012),237-247.

Page 7: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Distance-balanced graphs - example

Non-regular bipartite distance-balanced graph H.

Page 8: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Generalization: n-distance-balanced graphs

A graph Γ is said to be n-distance-balanced if there exist at leasttwo vertices at distance n in Γ and if for any two vertices u and vof Γ at distance n, the number of vertices closer to u than to v isequal to the number of vertices closer to v than to u.

We are intrested in 2-distance-balanced graphs.

Page 9: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Generalization: n-distance-balanced graphs

A graph Γ is said to be n-distance-balanced if there exist at leasttwo vertices at distance n in Γ and if for any two vertices u and vof Γ at distance n, the number of vertices closer to u than to v isequal to the number of vertices closer to v than to u.

We are intrested in 2-distance-balanced graphs.

Page 10: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

A distance partition of a graph with diameter 4 with respect tovertices u and v at distance 2.

Page 11: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs: example

Distance-balanced and 2-distance-balanced non-regular graph.

Page 12: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Question:Are there any 2-distance-balanced graphs that are notdistance-balanced?

Theorem (Handa, 1999):

Every distance-balanced graph is 2-connected.

A graph Γ is k-vertex-connected (or k-connected) if it has morethan k vertices and the result of deleting any set of fewer than kvertices is a connected graph.

Question:Are there any connected 2-distance-balanced graphs that are not2-connected?

Page 13: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Question:Are there any 2-distance-balanced graphs that are notdistance-balanced?

Theorem (Handa, 1999):

Every distance-balanced graph is 2-connected.

A graph Γ is k-vertex-connected (or k-connected) if it has morethan k vertices and the result of deleting any set of fewer than kvertices is a connected graph.

Question:Are there any connected 2-distance-balanced graphs that are not2-connected?

Page 14: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Question:Are there any 2-distance-balanced graphs that are notdistance-balanced?

Theorem (Handa, 1999):

Every distance-balanced graph is 2-connected.

A graph Γ is k-vertex-connected (or k-connected) if it has morethan k vertices and the result of deleting any set of fewer than kvertices is a connected graph.

Question:Are there any connected 2-distance-balanced graphs that are not2-connected?

Page 15: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Family of graphs Γ(G , c) - construction

Let G be an arbitrary graph (not necessary connected) and c anadditional vertex.

Then Γ(G , c) is a graph constructed in such a way that

V (Γ(G , c)) = V (G ) ∪ {c},

and

E (Γ(G , c)) = E (G ) ∪ {cv | v ∈ V (G )}.

I Γ(G , c) is connected.

I G is not connected ⇐⇒ Γ(G , c) is not 2-connected.

I Diameter of Γ = Γ(G , c) is at most 2.

Page 16: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Family of graphs Γ(G , c) - construction

Let G be an arbitrary graph (not necessary connected) and c anadditional vertex.

Then Γ(G , c) is a graph constructed in such a way that

V (Γ(G , c)) = V (G ) ∪ {c},

and

E (Γ(G , c)) = E (G ) ∪ {cv | v ∈ V (G )}.

I Γ(G , c) is connected.

I G is not connected ⇐⇒ Γ(G , c) is not 2-connected.

I Diameter of Γ = Γ(G , c) is at most 2.

Page 17: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Question:Are there any connected 2-distance-balanced graphs that are not2-connected?

Theorem (B.F., S. Miklavic)

Graph Γ is a connected 2-distance-balanced graph that is not2-connected iff Γ ∼= Γ(G , c) for some not connected regular graphG .

Page 18: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Question:Are there any connected 2-distance-balanced graphs that are not2-connected?

Theorem (B.F., S. Miklavic)

Graph Γ is a connected 2-distance-balanced graph that is not2-connected iff Γ ∼= Γ(G , c) for some not connected regular graphG .

Page 19: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Lemma (B.F., S. Miklavic)

If G is regular but not a complete graph, then Γ(G , c) is2-distance-balanced.

Proof:

Let G be a regular graph with valency k and construct Γ = Γ(G , c).

Let G1,G2, . . .Gn be connected components of G for some n ≥ 1.

Two essentially different types of vertices at distance two in Γ:

1. both from V (Gi ),

2. one from V (Gi ), the other from V (Gj) for i 6= j .

Page 20: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Lemma (B.F., S. Miklavic)

If G is regular but not a complete graph, then Γ(G , c) is2-distance-balanced.

Proof:

Let G be a regular graph with valency k and construct Γ = Γ(G , c).

Let G1,G2, . . .Gn be connected components of G for some n ≥ 1.

Two essentially different types of vertices at distance two in Γ:

1. both from V (Gi ),

2. one from V (Gi ), the other from V (Gj) for i 6= j .

Page 21: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Lemma (B.F., S. Miklavic)

If G is regular but not a complete graph, then Γ(G , c) is2-distance-balanced.

Proof:

Let G be a regular graph with valency k and construct Γ = Γ(G , c).

Let G1,G2, . . .Gn be connected components of G for some n ≥ 1.

Two essentially different types of vertices at distance two in Γ:

1. both from V (Gi ),

2. one from V (Gi ), the other from V (Gj) for i 6= j .

Page 22: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Lemma (B.F., S. Miklavic)

If G is regular but not a complete graph, then Γ(G , c) is2-distance-balanced.

Proof:

Let G be a regular graph with valency k and construct Γ = Γ(G , c).

Let G1,G2, . . .Gn be connected components of G for some n ≥ 1.

Two essentially different types of vertices at distance two in Γ:

1. both from V (Gi ),

2. one from V (Gi ), the other from V (Gj) for i 6= j .

Page 23: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof

1. Pick arbitrary v1, v2 ∈ V (Gi ) s.t. d(v1, v2) = 2.

W Γv1v2

= {v1} ∪ (NGi(v1) \ (NGi

(v1) ∩ NGi(v2)))

|W Γv1v2| = 1 + |NGi

(v1)| − |NGi(v1) ∩ NGi

(v2)|

|W Γv2v1| = 1 + |NGi

(v2)| − |NGi(v2) ∩ NGi

(v1)|

⇒ |W Γv1v2| = |W Γ

v2v1|

Page 24: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof

1. Pick arbitrary v1, v2 ∈ V (Gi ) s.t. d(v1, v2) = 2.

W Γv1v2

= {v1} ∪ (NGi(v1) \ (NGi

(v1) ∩ NGi(v2)))

|W Γv1v2| = 1 + |NGi

(v1)| − |NGi(v1) ∩ NGi

(v2)|

|W Γv2v1| = 1 + |NGi

(v2)| − |NGi(v2) ∩ NGi

(v1)|

⇒ |W Γv1v2| = |W Γ

v2v1|

Page 25: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof

1. Pick arbitrary v1, v2 ∈ V (Gi ) s.t. d(v1, v2) = 2.

W Γv1v2

= {v1} ∪ (NGi(v1) \ (NGi

(v1) ∩ NGi(v2)))

|W Γv1v2| = 1 + |NGi

(v1)| − |NGi(v1) ∩ NGi

(v2)|

|W Γv2v1| = 1 + |NGi

(v2)| − |NGi(v2) ∩ NGi

(v1)|

⇒ |W Γv1v2| = |W Γ

v2v1|

Page 26: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof

1. Pick arbitrary v1, v2 ∈ V (Gi ) s.t. d(v1, v2) = 2.

W Γv1v2

= {v1} ∪ (NGi(v1) \ (NGi

(v1) ∩ NGi(v2)))

|W Γv1v2| = 1 + |NGi

(v1)| − |NGi(v1) ∩ NGi

(v2)|

|W Γv2v1| = 1 + |NGi

(v2)| − |NGi(v2) ∩ NGi

(v1)|

⇒ |W Γv1v2| = |W Γ

v2v1|

Page 27: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof

1. Pick arbitrary v1, v2 ∈ V (Gi ) s.t. d(v1, v2) = 2.

W Γv1v2

= {v1} ∪ (NGi(v1) \ (NGi

(v1) ∩ NGi(v2)))

|W Γv1v2| = 1 + |NGi

(v1)| − |NGi(v1) ∩ NGi

(v2)|

|W Γv2v1| = 1 + |NGi

(v2)| − |NGi(v2) ∩ NGi

(v1)|

⇒ |W Γv1v2| = |W Γ

v2v1|

Page 28: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof

2. Pick arbitrary v ∈ V (Gi ), u ∈ V (Gj).

W Γvu = {v} ∪ NGi

(v)

W Γuv = {u} ∪ NGj

(u)

⇒ |W Γvu| = 1 + k = |W Γ

uv |

So Γ = Γ(G , c) is 2-distance-balanced.

Page 29: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof

2. Pick arbitrary v ∈ V (Gi ), u ∈ V (Gj).

W Γvu = {v} ∪ NGi

(v)

W Γuv = {u} ∪ NGj

(u)

⇒ |W Γvu| = 1 + k = |W Γ

uv |

So Γ = Γ(G , c) is 2-distance-balanced.

Page 30: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof

2. Pick arbitrary v ∈ V (Gi ), u ∈ V (Gj).

W Γvu = {v} ∪ NGi

(v)

W Γuv = {u} ∪ NGj

(u)

⇒ |W Γvu| = 1 + k = |W Γ

uv |

So Γ = Γ(G , c) is 2-distance-balanced.

Page 31: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof

2. Pick arbitrary v ∈ V (Gi ), u ∈ V (Gj).

W Γvu = {v} ∪ NGi

(v)

W Γuv = {u} ∪ NGj

(u)

⇒ |W Γvu| = 1 + k = |W Γ

uv |

So Γ = Γ(G , c) is 2-distance-balanced.

Page 32: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof

2. Pick arbitrary v ∈ V (Gi ), u ∈ V (Gj).

W Γvu = {v} ∪ NGi

(v)

W Γuv = {u} ∪ NGj

(u)

⇒ |W Γvu| = 1 + k = |W Γ

uv |

So Γ = Γ(G , c) is 2-distance-balanced.

Page 33: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Lemma (B.F., S. Miklavic)

If Γ is a connected 2-distance-balanced graph that is not2-connected then Γ ∼= Γ(G , c) for some not connected regulargraph G .

Proof:

Let Γ connected 2-distance-balanced graph that is not 2-connectedand c a cut vertex in Γ.

If we delete vertex c , we get some subgraph G with connectedcomponents G1,G2, . . .Gn for some n ≥ 2.

Page 34: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Lemma (B.F., S. Miklavic)

If Γ is a connected 2-distance-balanced graph that is not2-connected then Γ ∼= Γ(G , c) for some not connected regulargraph G .

Proof:

Let Γ connected 2-distance-balanced graph that is not 2-connectedand c a cut vertex in Γ.

If we delete vertex c , we get some subgraph G with connectedcomponents G1,G2, . . .Gn for some n ≥ 2.

Page 35: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Lemma (B.F., S. Miklavic)

If Γ is a connected 2-distance-balanced graph that is not2-connected then Γ ∼= Γ(G , c) for some not connected regulargraph G .

Proof:

Let Γ connected 2-distance-balanced graph that is not 2-connectedand c a cut vertex in Γ.

If we delete vertex c , we get some subgraph G with connectedcomponents G1,G2, . . .Gn for some n ≥ 2.

Page 36: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced graphs

Lemma (B.F., S. Miklavic)

If Γ is a connected 2-distance-balanced graph that is not2-connected then Γ ∼= Γ(G , c) for some not connected regulargraph G .

Proof:

Let Γ connected 2-distance-balanced graph that is not 2-connectedand c a cut vertex in Γ.

If we delete vertex c , we get some subgraph G with connectedcomponents G1,G2, . . .Gn for some n ≥ 2.

Page 37: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 1

Claim 1: c is adjacent to every vertex in Gl for at least one l ,1 ≤ l ≤ n.

Suppose this statement is not true. Then for arbitrary Gi and Gj :

∃v2 ∈ V (Gi ) s.t. d(c, v2) = 2

⇒ ∃v1 ∈ V (Gi ) s.t. d(c , v1) = d(v1, v2) = 1, and

∃u2 ∈ V (Gj) s.t. d(c, u) = 2

⇒ ∃u1 ∈ V (Gi ) s.t. d(c , u1) = d(u1, u2) = 1

Page 38: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 1

Claim 1: c is adjacent to every vertex in Gl for at least one l ,1 ≤ l ≤ n.

Suppose this statement is not true. Then for arbitrary Gi and Gj :

∃v2 ∈ V (Gi ) s.t. d(c, v2) = 2

⇒ ∃v1 ∈ V (Gi ) s.t. d(c , v1) = d(v1, v2) = 1, and

∃u2 ∈ V (Gj) s.t. d(c, u) = 2

⇒ ∃u1 ∈ V (Gi ) s.t. d(c , u1) = d(u1, u2) = 1

Page 39: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 1

W Γcv2⊇ {c} ∪ V (Gj)⇒ 1 + |V (Gj)| ≤ |W Γ

cv2|

W Γv2c ⊆ V (Gi ) \ {v1} ⇒ |W Γ

v2c | ≤ |V (Gi )| − 1

⇒ |V (Gj)| ≤ |V (Gi )| − 2

W Γcu2⊇ {c} ∪ V (Gi )⇒ 1 + |V (Gi )| ≤ |W Γ

cu2|

W Γu2c ⊆ V (Gj) \ {u1} ⇒ |W Γ

u2c | ≤ |V (Gj)| − 1

⇒ |V (Gi )|+ 2 ≤ |V (Gj)|

So: |V (Gi )|+ 2 ≤ |V (Gj)| ≤ |V (Gi )| − 2, contradiction.

From now on (w.l.o.g.): c is adjacent to every vertex in V (G1).

Page 40: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 1

W Γcv2⊇ {c} ∪ V (Gj)⇒ 1 + |V (Gj)| ≤ |W Γ

cv2|

W Γv2c ⊆ V (Gi ) \ {v1} ⇒ |W Γ

v2c | ≤ |V (Gi )| − 1

⇒ |V (Gj)| ≤ |V (Gi )| − 2

W Γcu2⊇ {c} ∪ V (Gi )⇒ 1 + |V (Gi )| ≤ |W Γ

cu2|

W Γu2c ⊆ V (Gj) \ {u1} ⇒ |W Γ

u2c | ≤ |V (Gj)| − 1

⇒ |V (Gi )|+ 2 ≤ |V (Gj)|

So: |V (Gi )|+ 2 ≤ |V (Gj)| ≤ |V (Gi )| − 2, contradiction.

From now on (w.l.o.g.): c is adjacent to every vertex in V (G1).

Page 41: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 1

W Γcv2⊇ {c} ∪ V (Gj)⇒ 1 + |V (Gj)| ≤ |W Γ

cv2|

W Γv2c ⊆ V (Gi ) \ {v1} ⇒ |W Γ

v2c | ≤ |V (Gi )| − 1

⇒ |V (Gj)| ≤ |V (Gi )| − 2

W Γcu2⊇ {c} ∪ V (Gi )⇒ 1 + |V (Gi )| ≤ |W Γ

cu2|

W Γu2c ⊆ V (Gj) \ {u1} ⇒ |W Γ

u2c | ≤ |V (Gj)| − 1

⇒ |V (Gi )|+ 2 ≤ |V (Gj)|

So: |V (Gi )|+ 2 ≤ |V (Gj)| ≤ |V (Gi )| − 2, contradiction.

From now on (w.l.o.g.): c is adjacent to every vertex in V (G1).

Page 42: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 1

W Γcv2⊇ {c} ∪ V (Gj)⇒ 1 + |V (Gj)| ≤ |W Γ

cv2|

W Γv2c ⊆ V (Gi ) \ {v1} ⇒ |W Γ

v2c | ≤ |V (Gi )| − 1

⇒ |V (Gj)| ≤ |V (Gi )| − 2

W Γcu2⊇ {c} ∪ V (Gi )⇒ 1 + |V (Gi )| ≤ |W Γ

cu2|

W Γu2c ⊆ V (Gj) \ {u1} ⇒ |W Γ

u2c | ≤ |V (Gj)| − 1

⇒ |V (Gi )|+ 2 ≤ |V (Gj)|

So: |V (Gi )|+ 2 ≤ |V (Gj)| ≤ |V (Gi )| − 2, contradiction.

From now on (w.l.o.g.): c is adjacent to every vertex in V (G1).

Page 43: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 1

W Γcv2⊇ {c} ∪ V (Gj)⇒ 1 + |V (Gj)| ≤ |W Γ

cv2|

W Γv2c ⊆ V (Gi ) \ {v1} ⇒ |W Γ

v2c | ≤ |V (Gi )| − 1

⇒ |V (Gj)| ≤ |V (Gi )| − 2

W Γcu2⊇ {c} ∪ V (Gi )⇒ 1 + |V (Gi )| ≤ |W Γ

cu2|

W Γu2c ⊆ V (Gj) \ {u1} ⇒ |W Γ

u2c | ≤ |V (Gj)| − 1

⇒ |V (Gi )|+ 2 ≤ |V (Gj)|

So: |V (Gi )|+ 2 ≤ |V (Gj)| ≤ |V (Gi )| − 2, contradiction.

From now on (w.l.o.g.): c is adjacent to every vertex in V (G1).

Page 44: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 1

W Γcv2⊇ {c} ∪ V (Gj)⇒ 1 + |V (Gj)| ≤ |W Γ

cv2|

W Γv2c ⊆ V (Gi ) \ {v1} ⇒ |W Γ

v2c | ≤ |V (Gi )| − 1

⇒ |V (Gj)| ≤ |V (Gi )| − 2

W Γcu2⊇ {c} ∪ V (Gi )⇒ 1 + |V (Gi )| ≤ |W Γ

cu2|

W Γu2c ⊆ V (Gj) \ {u1} ⇒ |W Γ

u2c | ≤ |V (Gj)| − 1

⇒ |V (Gi )|+ 2 ≤ |V (Gj)|

So: |V (Gi )|+ 2 ≤ |V (Gj)| ≤ |V (Gi )| − 2, contradiction.

From now on (w.l.o.g.): c is adjacent to every vertex in V (G1).

Page 45: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 1

W Γcv2⊇ {c} ∪ V (Gj)⇒ 1 + |V (Gj)| ≤ |W Γ

cv2|

W Γv2c ⊆ V (Gi ) \ {v1} ⇒ |W Γ

v2c | ≤ |V (Gi )| − 1

⇒ |V (Gj)| ≤ |V (Gi )| − 2

W Γcu2⊇ {c} ∪ V (Gi )⇒ 1 + |V (Gi )| ≤ |W Γ

cu2|

W Γu2c ⊆ V (Gj) \ {u1} ⇒ |W Γ

u2c | ≤ |V (Gj)| − 1

⇒ |V (Gi )|+ 2 ≤ |V (Gj)|

So: |V (Gi )|+ 2 ≤ |V (Gj)| ≤ |V (Gi )| − 2, contradiction.

From now on (w.l.o.g.): c is adjacent to every vertex in V (G1).

Page 46: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 1

W Γcv2⊇ {c} ∪ V (Gj)⇒ 1 + |V (Gj)| ≤ |W Γ

cv2|

W Γv2c ⊆ V (Gi ) \ {v1} ⇒ |W Γ

v2c | ≤ |V (Gi )| − 1

⇒ |V (Gj)| ≤ |V (Gi )| − 2

W Γcu2⊇ {c} ∪ V (Gi )⇒ 1 + |V (Gi )| ≤ |W Γ

cu2|

W Γu2c ⊆ V (Gj) \ {u1} ⇒ |W Γ

u2c | ≤ |V (Gj)| − 1

⇒ |V (Gi )|+ 2 ≤ |V (Gj)|

So: |V (Gi )|+ 2 ≤ |V (Gj)| ≤ |V (Gi )| − 2, contradiction.

From now on (w.l.o.g.): c is adjacent to every vertex in V (G1).

Page 47: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 2

Claim 2: Induced subgraph G1 is regular.

Observations: Pick arbitrary u ∈ V (G ) \ V (G1) adjacent to c .

I d(u, v) = 2 for every v ∈ V (G1)

I |W Γux | = |W Γ

uy | for arbitrary x , y ∈ V (G1)

I W Γvu = {v} ∪ (NΓ(v) \ {c}) for every v ∈ V (G1)

⇒ |W Γvu| = 1 + |NΓ(v)| − 1 = |NG1(v)|+ 1 for every

v ∈ V (G1)

⇒ |NG1(x)|+ 1 = |NΓ(x)| = |W Γxu| = |W Γ

ux | = |W Γuy | = |W Γ

yu| =|NΓ(y)| = |NG1(y)|+ 1 for arbitrary x , y ∈ V (G1)

From now on: Induced subgraph G1 is regular with valency k .(Every vertex in V (G1) has valency k + 1 in Γ.)

Page 48: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 2

Claim 2: Induced subgraph G1 is regular.

Observations: Pick arbitrary u ∈ V (G ) \ V (G1) adjacent to c .

I d(u, v) = 2 for every v ∈ V (G1)

I |W Γux | = |W Γ

uy | for arbitrary x , y ∈ V (G1)

I W Γvu = {v} ∪ (NΓ(v) \ {c}) for every v ∈ V (G1)

⇒ |W Γvu| = 1 + |NΓ(v)| − 1 = |NG1(v)|+ 1 for every

v ∈ V (G1)

⇒ |NG1(x)|+ 1 = |NΓ(x)| = |W Γxu| = |W Γ

ux | = |W Γuy | = |W Γ

yu| =|NΓ(y)| = |NG1(y)|+ 1 for arbitrary x , y ∈ V (G1)

From now on: Induced subgraph G1 is regular with valency k .(Every vertex in V (G1) has valency k + 1 in Γ.)

Page 49: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 2

Claim 2: Induced subgraph G1 is regular.

Observations: Pick arbitrary u ∈ V (G ) \ V (G1) adjacent to c .

I d(u, v) = 2 for every v ∈ V (G1)

I |W Γux | = |W Γ

uy | for arbitrary x , y ∈ V (G1)

I W Γvu = {v} ∪ (NΓ(v) \ {c}) for every v ∈ V (G1)

⇒ |W Γvu| = 1 + |NΓ(v)| − 1 = |NG1(v)|+ 1 for every

v ∈ V (G1)

⇒ |NG1(x)|+ 1 = |NΓ(x)| = |W Γxu| = |W Γ

ux | = |W Γuy | = |W Γ

yu| =|NΓ(y)| = |NG1(y)|+ 1 for arbitrary x , y ∈ V (G1)

From now on: Induced subgraph G1 is regular with valency k .(Every vertex in V (G1) has valency k + 1 in Γ.)

Page 50: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 2

Claim 2: Induced subgraph G1 is regular.

Observations: Pick arbitrary u ∈ V (G ) \ V (G1) adjacent to c .

I d(u, v) = 2 for every v ∈ V (G1)

I |W Γux | = |W Γ

uy | for arbitrary x , y ∈ V (G1)

I W Γvu = {v} ∪ (NΓ(v) \ {c}) for every v ∈ V (G1)

⇒ |W Γvu| = 1 + |NΓ(v)| − 1 = |NG1(v)|+ 1 for every

v ∈ V (G1)

⇒ |NG1(x)|+ 1 = |NΓ(x)| = |W Γxu| = |W Γ

ux | = |W Γuy | = |W Γ

yu| =|NΓ(y)| = |NG1(y)|+ 1 for arbitrary x , y ∈ V (G1)

From now on: Induced subgraph G1 is regular with valency k .(Every vertex in V (G1) has valency k + 1 in Γ.)

Page 51: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 2

Claim 2: Induced subgraph G1 is regular.

Observations: Pick arbitrary u ∈ V (G ) \ V (G1) adjacent to c .

I d(u, v) = 2 for every v ∈ V (G1)

I |W Γux | = |W Γ

uy | for arbitrary x , y ∈ V (G1)

I W Γvu = {v} ∪ (NΓ(v) \ {c}) for every v ∈ V (G1)

⇒ |W Γvu| = 1 + |NΓ(v)| − 1 = |NG1(v)|+ 1 for every

v ∈ V (G1)

⇒ |NG1(x)|+ 1 = |NΓ(x)| = |W Γxu| = |W Γ

ux | = |W Γuy | = |W Γ

yu| =|NΓ(y)| = |NG1(y)|+ 1 for arbitrary x , y ∈ V (G1)

From now on: Induced subgraph G1 is regular with valency k .(Every vertex in V (G1) has valency k + 1 in Γ.)

Page 52: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 2

Claim 2: Induced subgraph G1 is regular.

Observations: Pick arbitrary u ∈ V (G ) \ V (G1) adjacent to c .

I d(u, v) = 2 for every v ∈ V (G1)

I |W Γux | = |W Γ

uy | for arbitrary x , y ∈ V (G1)

I W Γvu = {v} ∪ (NΓ(v) \ {c}) for every v ∈ V (G1)

⇒ |W Γvu| = 1 + |NΓ(v)| − 1 = |NG1(v)|+ 1 for every

v ∈ V (G1)

⇒ |NG1(x)|+ 1 = |NΓ(x)| = |W Γxu| = |W Γ

ux | = |W Γuy | = |W Γ

yu| =|NΓ(y)| = |NG1(y)|+ 1 for arbitrary x , y ∈ V (G1)

From now on: Induced subgraph G1 is regular with valency k .(Every vertex in V (G1) has valency k + 1 in Γ.)

Page 53: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 2

Claim 2: Induced subgraph G1 is regular.

Observations: Pick arbitrary u ∈ V (G ) \ V (G1) adjacent to c .

I d(u, v) = 2 for every v ∈ V (G1)

I |W Γux | = |W Γ

uy | for arbitrary x , y ∈ V (G1)

I W Γvu = {v} ∪ (NΓ(v) \ {c}) for every v ∈ V (G1)

⇒ |W Γvu| = 1 + |NΓ(v)| − 1 = |NG1(v)|+ 1 for every

v ∈ V (G1)

⇒ |NG1(x)|+ 1 = |NΓ(x)| = |W Γxu| = |W Γ

ux | = |W Γuy | = |W Γ

yu| =|NΓ(y)| = |NG1(y)|+ 1 for arbitrary x , y ∈ V (G1)

From now on: Induced subgraph G1 is regular with valency k .(Every vertex in V (G1) has valency k + 1 in Γ.)

Page 54: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 2

Claim 2: Induced subgraph G1 is regular.

Observations: Pick arbitrary u ∈ V (G ) \ V (G1) adjacent to c .

I d(u, v) = 2 for every v ∈ V (G1)

I |W Γux | = |W Γ

uy | for arbitrary x , y ∈ V (G1)

I W Γvu = {v} ∪ (NΓ(v) \ {c}) for every v ∈ V (G1)

⇒ |W Γvu| = 1 + |NΓ(v)| − 1 = |NG1(v)|+ 1 for every

v ∈ V (G1)

⇒ |NG1(x)|+ 1 = |NΓ(x)| = |W Γxu| = |W Γ

ux | = |W Γuy | = |W Γ

yu| =|NΓ(y)| = |NG1(y)|+ 1 for arbitrary x , y ∈ V (G1)

From now on: Induced subgraph G1 is regular with valency k .(Every vertex in V (G1) has valency k + 1 in Γ.)

Page 55: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Claim 3: c is adjacent to every vertex in V (G ).

Suppose this statement is not true. Then:

∃u2 ∈ V (G2) s.t. d(c , u2) = 2

⇒ ∃u1 ∈ V (G2) s.t. d(c , u1) = (d(u1, u2) = 1

We know: |NΓ(v)| = k + 1 for arbitrary v in V (G1).

W Γvu1

= {v} ∪ (NΓ(v) \ {c})

|W Γvu1| = 1 + k + 1− 1 = k + 1

W Γcu2⊇ V (G1) ∪ {c}

|W Γcu2| ≥ |V (G1)|+ 1 ≥ k + 2

Page 56: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Claim 3: c is adjacent to every vertex in V (G ).

Suppose this statement is not true. Then:

∃u2 ∈ V (G2) s.t. d(c , u2) = 2

⇒ ∃u1 ∈ V (G2) s.t. d(c , u1) = (d(u1, u2) = 1

We know: |NΓ(v)| = k + 1 for arbitrary v in V (G1).

W Γvu1

= {v} ∪ (NΓ(v) \ {c})

|W Γvu1| = 1 + k + 1− 1 = k + 1

W Γcu2⊇ V (G1) ∪ {c}

|W Γcu2| ≥ |V (G1)|+ 1 ≥ k + 2

Page 57: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Claim 3: c is adjacent to every vertex in V (G ).

Suppose this statement is not true. Then:

∃u2 ∈ V (G2) s.t. d(c , u2) = 2

⇒ ∃u1 ∈ V (G2) s.t. d(c , u1) = (d(u1, u2) = 1

We know: |NΓ(v)| = k + 1 for arbitrary v in V (G1).

W Γvu1

= {v} ∪ (NΓ(v) \ {c})

|W Γvu1| = 1 + k + 1− 1 = k + 1

W Γcu2⊇ V (G1) ∪ {c}

|W Γcu2| ≥ |V (G1)|+ 1 ≥ k + 2

Page 58: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Claim 3: c is adjacent to every vertex in V (G ).

Suppose this statement is not true. Then:

∃u2 ∈ V (G2) s.t. d(c , u2) = 2

⇒ ∃u1 ∈ V (G2) s.t. d(c , u1) = (d(u1, u2) = 1

We know: |NΓ(v)| = k + 1 for arbitrary v in V (G1).

W Γvu1

= {v} ∪ (NΓ(v) \ {c})

|W Γvu1| = 1 + k + 1− 1 = k + 1

W Γcu2⊇ V (G1) ∪ {c}

|W Γcu2| ≥ |V (G1)|+ 1 ≥ k + 2

Page 59: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Claim 3: c is adjacent to every vertex in V (G ).

Suppose this statement is not true. Then:

∃u2 ∈ V (G2) s.t. d(c , u2) = 2

⇒ ∃u1 ∈ V (G2) s.t. d(c , u1) = (d(u1, u2) = 1

We know: |NΓ(v)| = k + 1 for arbitrary v in V (G1).

W Γvu1

= {v} ∪ (NΓ(v) \ {c})

|W Γvu1| = 1 + k + 1− 1 = k + 1

W Γcu2⊇ V (G1) ∪ {c}

|W Γcu2| ≥ |V (G1)|+ 1 ≥ k + 2

Page 60: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Claim 3: c is adjacent to every vertex in V (G ).

Suppose this statement is not true. Then:

∃u2 ∈ V (G2) s.t. d(c , u2) = 2

⇒ ∃u1 ∈ V (G2) s.t. d(c , u1) = (d(u1, u2) = 1

We know: |NΓ(v)| = k + 1 for arbitrary v in V (G1).

W Γvu1

= {v} ∪ (NΓ(v) \ {c})

|W Γvu1| = 1 + k + 1− 1 = k + 1

W Γcu2⊇ V (G1) ∪ {c}

|W Γcu2| ≥ |V (G1)|+ 1 ≥ k + 2

Page 61: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Define: U =⋃d

i=1 =(D i−1i ∪ D i

i

)W Γ

u2c ⊆ U ⊆W Γu1v

⇒ |W Γu2c | ≤ |W

Γu1v | = k + 1

⇒ k + 2 ≤ |W Γcu2| = |W Γ

u2c | ≤ k + 1, contradiction.

From now on: c is adjacent to every vertex in V (G ).

Page 62: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Define: U =⋃d

i=1 =(D i−1i ∪ D i

i

)W Γ

u2c ⊆ U ⊆W Γu1v

⇒ |W Γu2c | ≤ |W

Γu1v | = k + 1

⇒ k + 2 ≤ |W Γcu2| = |W Γ

u2c | ≤ k + 1, contradiction.

From now on: c is adjacent to every vertex in V (G ).

Page 63: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Define: U =⋃d

i=1 =(D i−1i ∪ D i

i

)W Γ

u2c ⊆ U ⊆W Γu1v

⇒ |W Γu2c | ≤ |W

Γu1v | = k + 1

⇒ k + 2 ≤ |W Γcu2| = |W Γ

u2c | ≤ k + 1, contradiction.

From now on: c is adjacent to every vertex in V (G ).

Page 64: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Define: U =⋃d

i=1 =(D i−1i ∪ D i

i

)W Γ

u2c ⊆ U ⊆W Γu1v

⇒ |W Γu2c | ≤ |W

Γu1v | = k + 1

⇒ k + 2 ≤ |W Γcu2| = |W Γ

u2c | ≤ k + 1, contradiction.

From now on: c is adjacent to every vertex in V (G ).

Page 65: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Define: U =⋃d

i=1 =(D i−1i ∪ D i

i

)W Γ

u2c ⊆ U ⊆W Γu1v

⇒ |W Γu2c | ≤ |W

Γu1v | = k + 1

⇒ k + 2 ≤ |W Γcu2| = |W Γ

u2c | ≤ k + 1, contradiction.

From now on: c is adjacent to every vertex in V (G ).

Page 66: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 3

Define: U =⋃d

i=1 =(D i−1i ∪ D i

i

)W Γ

u2c ⊆ U ⊆W Γu1v

⇒ |W Γu2c | ≤ |W

Γu1v | = k + 1

⇒ k + 2 ≤ |W Γcu2| = |W Γ

u2c | ≤ k + 1, contradiction.

From now on: c is adjacent to every vertex in V (G ).

Page 67: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 4

Claim 4: Graph G is regular with valency k.

W.l.o.g we prove that G2 is regular with valency k.

Pick arbitrary u ∈ V (G2) and v ∈ V (G1) (we already now:d(u, v) = 2).

W Γuv = {u} ∪ (NΓ(u) \ {c}) = {u} ∪ NG2(u)

W Γvu = {v} ∪ (NΓ(v) \ {c}) = {v} ∪ NG1(v)

⇒ |W Γuv | = 1 + |NG2(u)| in |W Γ

vu| = 1 + k .

Since |W Γuv | = |W Γ

vu|

⇒ |NG2(u)| = k for every u ∈ V (G2)�

Page 68: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 4

Claim 4: Graph G is regular with valency k.

W.l.o.g we prove that G2 is regular with valency k.

Pick arbitrary u ∈ V (G2) and v ∈ V (G1) (we already now:d(u, v) = 2).

W Γuv = {u} ∪ (NΓ(u) \ {c}) = {u} ∪ NG2(u)

W Γvu = {v} ∪ (NΓ(v) \ {c}) = {v} ∪ NG1(v)

⇒ |W Γuv | = 1 + |NG2(u)| in |W Γ

vu| = 1 + k .

Since |W Γuv | = |W Γ

vu|

⇒ |NG2(u)| = k for every u ∈ V (G2)�

Page 69: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 4

Claim 4: Graph G is regular with valency k.

W.l.o.g we prove that G2 is regular with valency k.

Pick arbitrary u ∈ V (G2) and v ∈ V (G1) (we already now:d(u, v) = 2).

W Γuv = {u} ∪ (NΓ(u) \ {c}) = {u} ∪ NG2(u)

W Γvu = {v} ∪ (NΓ(v) \ {c}) = {v} ∪ NG1(v)

⇒ |W Γuv | = 1 + |NG2(u)| in |W Γ

vu| = 1 + k .

Since |W Γuv | = |W Γ

vu|

⇒ |NG2(u)| = k for every u ∈ V (G2)�

Page 70: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 4

Claim 4: Graph G is regular with valency k.

W.l.o.g we prove that G2 is regular with valency k.

Pick arbitrary u ∈ V (G2) and v ∈ V (G1) (we already now:d(u, v) = 2).

W Γuv = {u} ∪ (NΓ(u) \ {c}) = {u} ∪ NG2(u)

W Γvu = {v} ∪ (NΓ(v) \ {c}) = {v} ∪ NG1(v)

⇒ |W Γuv | = 1 + |NG2(u)| in |W Γ

vu| = 1 + k .

Since |W Γuv | = |W Γ

vu|

⇒ |NG2(u)| = k for every u ∈ V (G2)�

Page 71: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 4

Claim 4: Graph G is regular with valency k.

W.l.o.g we prove that G2 is regular with valency k.

Pick arbitrary u ∈ V (G2) and v ∈ V (G1) (we already now:d(u, v) = 2).

W Γuv = {u} ∪ (NΓ(u) \ {c}) = {u} ∪ NG2(u)

W Γvu = {v} ∪ (NΓ(v) \ {c}) = {v} ∪ NG1(v)

⇒ |W Γuv | = 1 + |NG2(u)| in |W Γ

vu| = 1 + k .

Since |W Γuv | = |W Γ

vu|

⇒ |NG2(u)| = k for every u ∈ V (G2)�

Page 72: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 4

Claim 4: Graph G is regular with valency k.

W.l.o.g we prove that G2 is regular with valency k.

Pick arbitrary u ∈ V (G2) and v ∈ V (G1) (we already now:d(u, v) = 2).

W Γuv = {u} ∪ (NΓ(u) \ {c}) = {u} ∪ NG2(u)

W Γvu = {v} ∪ (NΓ(v) \ {c}) = {v} ∪ NG1(v)

⇒ |W Γuv | = 1 + |NG2(u)| in |W Γ

vu| = 1 + k .

Since |W Γuv | = |W Γ

vu|

⇒ |NG2(u)| = k for every u ∈ V (G2)�

Page 73: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

Proof - Claim 4

Claim 4: Graph G is regular with valency k.

W.l.o.g we prove that G2 is regular with valency k.

Pick arbitrary u ∈ V (G2) and v ∈ V (G1) (we already now:d(u, v) = 2).

W Γuv = {u} ∪ (NΓ(u) \ {c}) = {u} ∪ NG2(u)

W Γvu = {v} ∪ (NΓ(v) \ {c}) = {v} ∪ NG1(v)

⇒ |W Γuv | = 1 + |NG2(u)| in |W Γ

vu| = 1 + k .

Since |W Γuv | = |W Γ

vu|

⇒ |NG2(u)| = k for every u ∈ V (G2)�

Page 74: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced cartesian product G�H

Question:Which cartesian products G�H are 2-distance-balanced?

Cartesian product of graphs G and H, denoted by G�H, is agraph with vertex set V (G )× V (H), where (u1, v1) and (u2, v2)are adjacent if and only if either

I u1 = u2 and v1 is adjacent to v2 in H, or

I v1 = v2 and u1 is adjacent to u2 in G .

Page 75: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced cartesian product G�H

Question:Which cartesian products G�H are 2-distance-balanced?

Cartesian product of graphs G and H, denoted by G�H, is agraph with vertex set V (G )× V (H), where (u1, v1) and (u2, v2)are adjacent if and only if either

I u1 = u2 and v1 is adjacent to v2 in H, or

I v1 = v2 and u1 is adjacent to u2 in G .

Page 76: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced cartesian product G�H

Theorem (B.F., S. Miklavic)

Cartesian product G�H is 2-distance-balanced iff one of thefollowing statements is true:

(i) Both graphs G an H are 2-distance-balanced and1-distance-balanced.

(ii) G is a complete graph Kn for some n ≥ 2 and H is aconnected 2-distance-balanced and 1-distance-balanced graph.

(iii) H is a complete graph Kn for some n ≥ 2 and G is aconnected 2-distance-balanced and 1-distance-balanced graph.

(iv) G is a complete graph Kn and H is a complete graph Km forsome m, n ≥ 2.

Page 77: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced lexicographic product G [H]

Question:Which lexicographic products G [H] are 2-distance-balanced?

Lexicographic product of graphs G and H, denoted by G [H], is agraph with vertex set V (G )× V (H), where (u1, v1) and (u2, v2)are adjacent if and only if either

I u1 is adjacent to u2 in G , or

I u1 = u2 and v1 is adjacent to v2 in H.

Page 78: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced lexicographic product G [H]

Question:Which lexicographic products G [H] are 2-distance-balanced?

Lexicographic product of graphs G and H, denoted by G [H], is agraph with vertex set V (G )× V (H), where (u1, v1) and (u2, v2)are adjacent if and only if either

I u1 is adjacent to u2 in G , or

I u1 = u2 and v1 is adjacent to v2 in H.

Page 79: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

2-distance-balanced lexicographic product G [H]

Theorem (B.F., S. Miklavic)

Lexicographic product G [H] is 2-distance-balanced iff one of thefollowing statements is true:

(i) G is a connected 2-distance-balanced graph and H is a regulargraph.

(ii) G is a complete graph and H is a regular graph, which is notcomplete.

(iii) G is a complete graph and H is a connected completebipartite graph.

Page 80: 2-DISTANCE-BALANCED GRAPHS · Motivation: Distance-balanced graphs IThese graphs were rst studied (at least implicitly) by K. Handa who considered distance-balanced partial cubes.

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