Weakly Connected Domination Koh Khee Meng Department of Maths National U of Singapore...

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Transcript of Weakly Connected Domination Koh Khee Meng Department of Maths National U of Singapore...

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  • Weakly Connected Domination Koh Khee Meng Department of Maths National U of Singapore matkohkm@nus.edu.sg
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  • Three Related Dominations V S
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  • domination number The domination number of G = (G) = min{| S | : S is a ds in G}. -set Call a minimum ds of G a -set of G. [1977] 1 st survey paper by Cockayne & Hedetniemi
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  • connected ds A ds S is called a connected ds if the induced subgraph [S] of G is connected. [1979] Sampathkumar & Walikar
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  • Disjoint union of two K(1, 3)s
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  • weakly connected ds wcds A ds S of G is a weakly connected ds (wcds) of G if is connected. Connected
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  • weakly connected domination The weakly connected domination was first introduced by Grossman (1997)
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  • Relations with other parameters
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  • Sanchis works
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  • Trees
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  • Cycle-e-disjoint Graph Cactus Cycle-e-disjoint Graph (Cactus)
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  • Lemanska (2007) Koh & Xu (2008) Extended the above to unicyclic graphs.
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  • cactus Let G be a cactus of order n 3; z(G) = number of end-vertices, c(G) = number of cycles, oc(G) = number of odd cycles in G.
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  • Cacti Cacti for which equality holds are characterized.
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  • Applications Mobile Ad-hoc Networks
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  • Vizings Conjecture (1968)
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  • Thank You !
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  • Applications
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