Weakly Connected Domination

1. Weakly Connected Domination Koh Khee Meng Department of Maths National U of Singapore matkohkm@nus.edu.sg

2. Three Related Dominations Let G = (V, E) be a connected graph. For , N(v) = the set of neighbors of v, N[v] = N(v) For , ] = • S : dominating set(ds) if i.e., every vertex in V \ S is adjacent to a vertex in S. V S

3. • The domination numberof G = γ(G) = min{| S | : Sis a ds in G}. Call a minimumdsof G a γ-setof G. [1977] 1st survey paper by Cockayne & Hedetniemi The n-cube graph,n = 3

4. • Ads S is called a connectedds if the induced subgraph [S] of G is connected. • The connected domination number of G = (G) = min{| S | : Sis a connectedds in G}. Call a minimum connected dsof G a -setof G. ≤ = 4 [1979] Sampathkumar & Walikar

5. The subgraphweakly induced by S = ( Disjoint union of two K(1, 3)’s

6. A dsS of G is a weakly connected ds (wcds) of G if is connected. The weakly connected domination number of G = (G) = min{| S | : Sis a wcdsin G}. Call a minimum wcdsof G a -setof G. () = n ? = 3 Connected

7. 4 •() = k ≤

8. The weakly connected domination was first introduced by Grossman (1997) ● The problem of computing is NP-hard in general.

9. • ≤ 2

10. and • ≤ 2

11. Relations with other parameters Notation = the independence number of G, = the vertex-covering number of G. Let G be a connected graph of order n ≥2. Then • ≤ n/2. • ≤ = () = k

12. = the connectivity of G. Let G be a connected graph of order n ≥ 2. Then ≤ n - The equality holds iff or (p-partite, n = 2p).

13. Sanchis’ works [1991]Let G be a graph of order n and Then e(G) ≤ . [2000] Let G be a graph of order n and Then e(G) ≤ + (k – 1). Grossman asked : How about e(G) if = k?

14. Let G be a graph of order n and Then e(G) ≤ . The extremal graphs are characterized.

15. Trees If T is a tree of order n ≥ 2, then = = The problem of computing is linear for trees.

16. Let G be a graph. Then = G has a -set Ghas a Ss.t. u They provide a constructive characterization of treesT for which =.

17. LetTbea tree of order n ≥ 3 and z(T) the number of end-vertices in T. Then [2004] ≥ (

18. LetTbea tree of order n ≥ 3 and z(T) the number of end-vertices in T. Then •≥ ( ≥ ( Letbe the family of trees defined recursively as follows:

19. ( = (21 – 8 + 1)/2 = 7 = the family of trees Ts.t. =(

20. Cycle-e-disjoint Graph (Cactus) connected graph G is acactus if no two cycles in G have an edge in common; unicyclic if it has exactly one cycle.

21. Lemanska (2007) For treeT, •≥ ( Koh & Xu (2008) Extended the above to unicyclic graphs.

22. Let G be a cactus of order n ≥ 3; z(G) = number of end-vertices, c(G) = number of cycles, oc(G) = number of oddcycles in G. Then ≥ ½ RHS = ½(14 – 2 + 1 – 3 – 2) = 4 < 5 =

23. ½ = ½(14 – 1 + 1 – 3 – 1) = 5 = Cactifor which equality holds are characterized.

24. A graph G is -stable if (G+e) = (G) edge e in . For a tree T, TFAE: (1) T is -stable; (2) there is a uniquemaximum independent set in T; (3)there is a unique-set in T.

25. G is -unique if it has a unique -set. G is cycle-disjoint if no 2 cycles in G have a vertex in common. The family of -unique cycle-disjointgraphs is completely determined.

27. Applications Mobile Ad-hoc Networks

28. Problems • Study (G ) and (G ). Vizing’s Conjecture (1968)

29. • Let G be a connected graph in which every block is either a a cycle or a cycle with a chord. Study (G). • Study the criticalityof graphs wrt.

30. Thank You !

31. Applications