A Linear Vizing-likeRelation Between the Sizeand the DominationNumber of a Graph

Dieter RautenbachLEHRSTUHL II FUR MATHEMATIK

RWTH-AACHEN, GERMANYE-mail: RAUTEN@MATH2.RWTH-AACHEN.DE

Received July 3, 1997; accepted May 15, 1998

Abstract: We prove m ≤ ∆n − (∆ + 1)γ for every graph without isolatedvertices of order n, size m, domination number γ and maximum degree ∆ ≥ 3.This generalizes a result of Fisher et al., CU-Denver Tech Rep, 1996] who obtainedthe given bound for the case ∆ = 3. c© 1999 John Wiley & Sons, Inc. J Graph Theory 31: 297–302,

1999

1. INTRODUCTION

All graphs in this article are finite, undirected, and contain no loops or multipleedges. For any graph G, the vertex set and edge set are denoted by V (G) andE(G). The order |V (G)| and the size |E(G)| ofG are denoted by n(G) andm(G).The neighborhood of a vertex x ∈ V (G) in the graph G is denoted by N(x,G),and the closed neighborhood {x} ∪N(x,G) by N [x,G]. For any set D ⊂ V (G)we define N(D,G) = ∪x∈DN(x,G) and N [D,G] = ∪x∈DN [x,G]. For anytwo subsets A,B ⊂ V (G), the graph G[A,B] has vertex set A ∪ B and containsall edges of G joining one vertex in A and one vertex in B. Furthermore, wedefine G[A] = G[A,A] and G − A = G[V (G) − A]. The graph G + ab fora, b ∈ V (G) has vertex set V (G) and edge set E(G + ab) = E(G) ∪ {ab}. We

c© 1999 John Wiley & Sons, Inc. CCC 0364-9024/99/030297-06

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write G = H ◦K1 if V (G) = {x, x′|x ∈ V (H)}, and E(G) = E(H) ∪ {xx′|x ∈V (H)} for two graphs H and G. A set D ⊂ V (G) is a dominating set of G iffN [D,G] = V (G). The minimum cardinality of a dominating set is the dominationnumber γ(G) of G. The degree d(x,G) of a vertex x ∈ V (G) is |N(x,G)|. ∆(G)and δ(G) denote the maximum and minimum degree of a graph G. If d(x,G) = rfor all vertices x ∈ V (G) and some integer r, then G is called r-regular. When-ever it is clear to which graph we refer, we write d(x), N(x), . . . etc. instead ofd(x,G), N(x,G), . . . etc.

A classical result of Vizing [6] relates the size m and the domination number γof a graph of order n:

m ≤ b(n− γ)(n− γ + 2)/2c.This bound on m is tight, but the graphs for which equality is achieved have twoextreme properties. First, for γ ≥ 2 they are disconnected, second, their edges arevery unevenly distributed, i.e., ∆(G) and δ(G) differ very much (δ(G) = 0 and∆(G) = n− γ).

In [4] and [5], Sanchis offered analogous results for connected graphs, but theedges of the extremal graphs are still very unevenly distributed.

An easy condition on a graph G, which implies that its edges are in some senseevenly distributed, is to limit its maximum degree ∆(G).

Very recently, Fisher, Fraughnaugh, and Seager [2] found a tight result (seeCorollary 1), which can be interpreted as an attempt to find the appropriate de-pendence of m on n and γ, if the maximum degree ∆ is bounded from aboveby three.

In Section 2, we prove a bound on m in terms of n,∆, and γ for general valuesof ∆, which generalizes this result of Fisher, Fraughnaugh, and Seager. A part ofour proof relies on a result of Reed [3].

Theorem 1 (B. Reed [3]). Let G be a graph of order n with minimum degreeδ(G) ≥ 3, then

γ(G) ≤ 38n.

The square dependence on n and γ in Vizing’s result turns into a linear de-pendence on n, γ, and ∆, if we demand a more even distribution of the edges byrestricting the maximum degree. This is rather intuitive, since in some sense thefactor ∼ (n− γ), which corresponded to the maximum degree, is replaced by thefactor ∼ ∆.

We believe that our result can still be improved if ∆ ≥ 4, and offer a correspond-ing conjecture.

2. BOUND

Theorem 2. Let G = (V,E) be a graph on n vertices without isolated verticesand ∆ ≥ 3 an integer. If G has maximum degree ∆(G) ≤ ∆, domination number

A VIZING-LIKE RELATION 299

γ and m edges, then

m ≤ ∆n− (∆ + 1)γ. (1)

Proof. We proceed by induction on n. For n = 2, the result is trivial. Let G bea graph on n > 2 vertices without isolated vertices. If ∆(G) ≤ 2, then the theoremis easily verified (note that in this case ∆(G) < ∆). Since the right side of (1) ismonotonically increasing in ∆, we can assume that

∆ = ∆(G) ≥ 3. (2)

The same reason together with the fact that (1) is linear inm,n, andγ implies that wecan assume that G is connected. Let y ∈ V be a vertex with d(y) = δ = δ(G) andx ∈ N(y). If S is the set of isolated vertices inG−N [x], thenG′ = G−N [x]−Shas no isolated vertices.

We set s = |S| and G′′ = G[N [x] ∪ S]. If m1 is the number of edges of Gincident with vertices in N(x), then m = m1 + m(G′) and m1 ≤ ∆(d(x) − 1)+ δ. Since n = n(G′) + d(x) + s + 1 and γ ≤ γ(G′) + γ(G′′), we obtain byinduction

m ≤ m1 + ∆(n− d(x)− s− 1)− (∆ + 1)(γ − γ(G′′)) (3)

= ∆n− (∆ + 1)γ +m1 −∆(d(x) + s+ 1) + (∆ + 1)γ(G′′) (4)

≤ ∆n− (∆ + 1)γ − 2∆ + δ −∆s+ (∆ + 1)γ(G′′). (5)

Inequality (4) or (5) implies (1), if either

m1 −∆(d(x) + s+ 1) + (∆ + 1)γ(G′′) ≤ 0 (6)

or

−2∆ + δ −∆s+ (∆ + 1)γ(G′′) ≤ 0. (7)

If s ≤ ∆− δ− 1 or s ≥ (∆+1)d(x)+δ−2∆∆ , then we obtain (7), since γ(G′′) ≤ s+ 1

and γ(G′′) ≤ d(x) (the sets N(x) and S ∪ {x} are dominating sets of G′′). Thus,we assume

∆− δ ≤ s < (∆ + 1)d(x) + δ − 2∆∆

≤ d(x) ≤ ∆, (8)

which implies that d(x) > ∆− δ + 1− 1∆+1 or, equivalently,

d(x) ≥ ∆− δ + 1. (9)

If there is a vertex z ∈ N(x) that has two neighbors a, b in S, then γ(G′′) ≤ s(the set {x, z} ∪ (S − {a, b}) is a dominating set of G′′) and (7) holds. Hence, weassume furthermore that every vertex in N(x) has at most one neighbor in S. Weconsider all possible cases.Case 1. δ ≥ 3.

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We have m ≤ ∆2 n and γ ≤ 3

8n, by Theorem 1. Thus, by (2), we obtain

m ≤ ∆2n ≤ ∆n− (∆ + 1)

38n

≤ ∆n− (∆ + 1)γ.

Case 2. δ = 1.By (8) and (9), we have d(x) = ∆ and ∆− 1 ≤ s < ∆− 1 + 1

∆ , which impliess = ∆− 1.

This implies d(u) = 1 for all u ∈ S, and every vertex z ∈ N(x) − {y} hasexactly one neighbor in S. If there is a vertex z ∈ N(x)−{y} with d(z) ≤ ∆− 1,then m1 ≤ ∆(∆− 1) + δ − 1 and, using γ(G′′) ≤ ∆, we obtain (1) by verifyingthat (6) holds:

m1 −∆(∆ + ∆) + (∆ + 1)∆ ≤ 0.

Thus, we assume that d(z) = ∆ for all z ∈ N(x)− {y}.As G is connected, y was an arbitrary vertex of degree 1, and x was the unique

neighbor of y, we can now deduce that G = H ◦K1 for a (∆ − 1)-regular graphH . For such a graph, m = ∆+1

4 n and γ = 12n and using (2) we obtain (1).

Case 3. δ = 2.By (8), there are at least 2s ≥ 2∆− 4 edges between S and N(x). Since every

vertex in N(x) has at most one neighbor in S, this implies

∆ ≥ d(x) ≥ 2∆− 4,

and, hence, ∆ ≤ 4.Case 3.1. ∆ = 4.

By (8) and (9), we have d(x) ∈ {3, 4} and s ∈ {2, 3}. If d(x) = 3 or s = 3 ors = 2, and there is a vertex a ∈ S with d(a) ≥ 3, then some vertex in N(x) hastwo neighbors in S, which is a contradiction. Hence, d(x) = 4, all s = 2 verticesin S have degree 2, and every vertex in N(x) is adjacent to exactly one vertexin S.

If u is the neighbor of y in S and y′ 6= y is the second neighbor of u, thenG1 = G− {y, y′, u} has no isolated vertices and, with γ ≤ γ(G1) + 1, we obtain(1) by induction:

m ≤ 6 + 4(n− 3)− 5(γ − 1) < 4n− 5γ.

Case 3.2. ∆ = 3.By (9), we have d(x) ∈ {2, 3}. If d(x) = 2, then (8) implies s = 1. AsG 6= C4,

we obtain N(x) = {y, y′}, S = {s′}, N(s′) = {y, y′}, and d(y′) = 3. Since thegraph G2 = G − {x, y, s′} contains no isolated vertices and γ ≤ γ(G2) + 1, weobtain (1) by induction:

m ≤ 4 + 3(n− 3)− 4(γ − 1) < 3n− 4γ.

A VIZING-LIKE RELATION 301

Hence, d(x) = 3 and, by (8), s ∈ {1, 2}. If s = 2, then there is a vertex in N(x)that has two neighbors in S, which is a contradiction. Hence, s = 1 and we haveN(x) = {y, y′, y′′} and S = {u}.

If two vertices in N(x) are adjacent or either d(y′) = 2 or d(y′′) = 2, thenm1 ≤ 7. Thus, we obtain (1) by verification of (6):

7− 3(3 + 1 + 1) + 4(1 + 1) = 0.

Therefore, we assume that d(y′) = d(y′′) = 3 and N(x) is independent.We now choose two vertices a, b from N(x). If u is adjacent to y, then a = y

and b is a second neighbor of u. If u is not adjacent to y, then a = y′, b = y′′. Fora neighbor c of b in V (G)−N [x]− S, we consider the graph G3 = G− {a, b, u}+ xc without isolated vertices. Let D3 be a dominating set of G3. If x ∈ D3, thenD3 ∪ {b} is a dominating set of G, if c ∈ D3, then D3 ∪ {a} is a dominating setof G, and if, finally, c, x 6∈ D3, then D3 ∪ {u} is a dominating set of G. Hence,γ ≤ γ(G3) + 1. Furthermore, we have m − m(G3) ≤ 3 + 3 − 1 = 5 and weobtain (1) by induction:

m ≤ 5 + 3(n− 3)− 4(γ − 1) = 3n− 4γ.

It is now easy to deduce the result of [2].

Corollary 1 (Fisher, Fraughnaugh, and Seager [2]). Let G be a graph of ordern, size m, domination number γ, maximum degree ∆ ≤ 3, and i isolated vertices,then γ ≤ 1

4(3n−m+ i).

In order to show the sharpness of their result, Fisher, Fraughnaugh, and Seagercite the graphs Cl ◦K1 (where Cl is a cycle on l ≥ 3 vertices) and six other graphswithout isolated vertices.

By a tedious case analysis, we are able to show that these graphs are exactly allconnected graphs without isolated vertices, which achieve equality in (1) for thecase ∆ = 3.

Thus, for ∆ = 3, Theorem 2 is sharp, and we can characterize all extremalgraphs. Fisher, Fraughnaugh, and Seager use a different approach in their proof ofCorollary 1, which is, therefore, a bit lengthy (about 14 pages). They consider alarge number of different cases according to the minimum degree δ of the graph.Their approach appears to be inconvenient to deal with graphs of maximum degree∆ ≥ 4, since the number of cases they have to take into account seems to growexponentially with ∆. On the other hand, their proof is independent of the result ofReed, which we use in our proof.

In view of the trivial boundm(G) ≤ ∆(G)2 n(G) for every graph, we believe that

Theorem 2 is not tight for large ∆ and conjecture the following.

Conjecture 1. LetG = (V,E) be a graph on n vertices without isolated verticesand ∆ ≥ 3 an integer. If G has maximum degree ∆(G) ≤ ∆, domination numberγ and m edges, then m ≤ ∆+3

2 n− ∆+52 γ.

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There are some points in favor of this conjecture. First, it would be tight forthe graphs H ◦ K1 for (∆ − 1)-regular graphs H . These graphs appear ratheroften as extremal graphs for domination matters. Second, our proof forced us tosacrifice a factor 1

2 , which is contained in the Vizing bound but not in Theorem 2.In Conjecture 1, this factor is recovered. A disadvantage of Conjecture 1 is that for∆-regular graphs it implies γ ≤ 3

∆+5n, which has not the correct asymptotics for

large ∆, which would be ∼ log(∆+1)∆+1 n [1].

ACKNOWLEDGMENTS

I would like to thank the anonymous referees for their interest and very goodremarks, which helped to shorten the proof.

References

[1] V. I. Arnautov, Estimation of the external stability number of a graph by meansof the minimum degree of the vertices, Prikl Mat i Prog Vyp 11 (1974), 3–8.

[2] D. Fisher, K. Fraughnaugh, and S. Seager, Domination in graphs with maxi-mum degree three, CU-Denver Tech Rep, 1996.

[3] B. Reed, Paths, stars, and the number three, Comb Prob Comp 5 (1996),277–295.

[4] L. A. Sanchis, Maximum number of edges in connected graphs with a givendomination number, Disc Math 87 (1991), 65–72.

[5] L. A. Sanchis, Some results on characterizing the edges of connected graphswith a given domination number, Disc Math 140 (1995), 149–166.

[6] V. G. Vizing, An estimate on the external stability number of a graph, DoklAkad Nauk SSSR 164 (1965), 729–731.