Dominating Pair Graphs

1. Dominating Pair Graphs Jitender S. Deogun and Dieter Kratsch SIAM J. Discrete Math. Vol.15(3) pp. 353-366 Liao Chung-Shou July, 2003

2. TOPIC • Introduction • Preliminaries • Weak dominating pair graphs • A polar theorem • Path MCDS (minimum connected dominating set)

3. Introduction --- main definition • dominating pair (definition) • A pair of vertices of a graph is called a dominating pair if the vertex set of every path between these two vertices is a dominating set of the graph. • dominating pair graphs (definition) • each of its connected induced subgraph has a dominating pair. • diametral path graphs (definition) • each of its connected induced subgraph has a dominating diametral path. (a path whose length is diameter).

4. dominating pair graphs Perfect graphs AT-free graphs chordal graphs cocomparability graphs trapezoid graphs strongly chordal graphs permutation graphs interval graphs trees Introduction --- graph class relationship

5. Preliminaries --- basic G=(V,E): |V|=n, |E|=m. G[W]: subgraph induced by W, where W  V. N(x)={v : xv  E}, N[x]=N(x)  {x}. dominating set D : D  V and each vertex in V\D has a neighbor in D. minimum dominating set D of G : |D| = (G). a simple path P=(u=x0, x1, …, xn=v), where xixj. P is called u.v-path. If G[V(P)]  Pn+1 , then P is called induced path (or chordless path).

6. Preliminaries --- about our content 1 • diametral path graphs : each of its connected induced subgraph has a • dominating diametral path. (a path whose length is diameter). • different : • diametral dominating pair (diametral DP) : a dominating pair and a diametral path. • AT-free graph : • AT : three indep. vertices x, y, z in V(G) is called asteroidal tripe (AT) if for any • two vertices of these three, there exists a path joining them and avoids the • closed neighborhood of the third. • AT-free graph : A graph doesn’t contain AT.

7. x b e f h a c g d z y x DP graph \ AT-free graph y z Not every DP graph has diametral DP i.e. no DP of length 4. Preliminaries --- about our content 2 Theorem: (point out the relationship of AT-free graphs & DP graphs) Any connected AT-free graph has a DP. Moreover, any connected AT-free graph has a diametral DP.

8. Weak dominating pair graphs • weak dominating pair graphs • A graph G is a weak DP graph if G has a DP. • minimal x,y-separator • G=(V,E) is a graph and SV is a separator of G if G[V\S] is disconnected. S is x,y-separator if x and y are in distinct components in G[V\S].(minimal:no proper separator contained in S) • far component • G=(V,E) is a graph and S is a separator of G. A component C of G[V\S] is a far component if there is a vertex of C that is not adjacent to every vertex of S.

9. G Cy Cx S every x,y-path pass through S The properties of weak DP graphs Theorem: Let S be a minimal x,y-separator of G and Cx and Cy be the components of G[V\S] containing x and y. Then every vertex of S has one neighbor in Cx and one in Cy. Lemma: G=(V,E) and a DP x, y in G that xyE. S is minimal x,y-separator of G. The only possible far components of G[V\S] are Cx, Cy. <proof>: a vertex z in C Cx, Cy, such that zsE, sS. Then sS s.t. x,s-path in Cx and s,y-path in Cy avoid N[z]. Thus, x,y-path doesn’t dominates z. A contradiction.

10. x z y C7 has 2-AT. The properties of weak DP graphs Lemma : G=(V,E) is a weak DP graph. G cannot have a 3-distant AT. (k-distant AT : the distance of each pair of AT  k) The inverse is wrong. <proof> : Let x, y be DP and {a,b,c} be 3-distant AT. Let x,y-path=(x=x0, x1,…, xt=y) and w=xi is the leftmost vertex adjacent to {a,b,c} and z=xj is the rightmost …… First, no two of {a,b,c} have common neighbors, since 3-distant AT. Second, assume one of {a,b,c}, say a, aw, az  E. Then a path (x=x0, x1,…, w=xi, a, xj=z,…,xt=y) avoid N[b] and N[c], a contradiction. Third, a is unique in {a,b,c} adjacent to w and c is unique in {a,b,c} adjacent to z. Then  a,c-path avoid N[b] and  a path (x,…w,a,…,c,z,…,y) avoid N[b].

11. A polar theorem --- statement & proof Theorem : Let G=(V,E) be a weak dominating pair graph of diameter at least 5. Then there are disjoint sets XV and YV such that for all x, y  V, x, y is a dominating pair of G if and only if x  X and y Y. The idea of the proof : Assume x, y and x’, y’ are two DP, then 利用反證法we claim that there is an x,y’-path (or x’,y-path) P of G such that V \ N[P]  .

12. e g b h b e f h j a a c g A counterexample of a DP graph of diameter=4. DP : (a,g), (b,g), (c,f), (c,h), (c,g). c d i d f Another question about diametral DP : each weak DP graph of diameter  5 has a diametral DP ? A polar theorem --- remark edge eg and df can extend……

13. Path MCDS --- introduction • Definition of connected dominating set: • A dominating set D of G is called a connected dominating set if G[D] is connected. (minimum size : conn(G)) • Definition of Path-MCDS: • D is a Path-MCDS if D is a MCDS and G[D]  P|D|. • Previous results and our goal: • each diametral path graph of diameter  5 has a Path-MCDS. • each dominating pair graphs has a Path-MCDS. (Goal ?)

14. Path MCDS --- diameter 1,2,4 proof Lemma : If G=(V,E) has a dominating pair that is not diametral, then G has a Path-MCDS. <proof> : DP x, y and we know dG(x,y)  diam(G)1. Let P be a shortest x,y-path. Besides, conn(G)  diam(G)1=|diameter path|2. (so diam(G)1  |P|  diam(G)) 1. if |P| = diam(G)1, it’s OK. 2. if |P| = diam(G) and conn(G)=diam(G)1, then since any MCDS contains diam(G)1 internal vertices in a diameter. Lemma : G=(V,E) has a diametral DP x, y, then 1. if conn(G)  diam(G), then G has a Path-MCDS. 2. if conn(G) = diam(G) and D is a MCDS of G that G[D] isn’t a path, then x, y D, and dG[D](a,b)=diam(G)2 for all aN(x)D and bN(y)D.

15. paw claw diamond C4 G has a diametral DP x, y, and conn(G)  diam(G), then G has a Path-MCDS. <proof> : assume P is a shortest x,y-path and |P|=k+1. Then dG(x,y)=k=diam(G). Let D be MCDS, k1  |D|=conn(G)  k+1. If conn(G) = k+1, then P is Path-MCDS. If conn(G) = k 1, then D is Path-MCDS since if diam(G[D])<k2, a contradiction. G=(V,E) has a diametral DP x, y, and conn(G) = diam(G) and D is a MCDS of G that G[D] isn’t a path, then x, y D, and dG[D](a,b)=diam(G)2 for all aN(x)D and bN(y)D. Every dominating pair graph G of diameter{1,2,4} has a Path-MCDS. <反證法> : only need to consider diam(G)=4. (diam(G[D])  2 for any MCDSD)

16. Path MCDS --- diameter  5 proof Every connected dominating pair graph G of diameter  5 has a Path-MCDS. <proof> : 反證法 G has a diametral DP x, y and a MCDS D that |D| = conn(G) = diam(G)=k  5. By lemma, x, y D, and dG[D](a,b)=k2 for all aN(x)D and bN(y)D. Let a path P=(a=a1, a2,…, ak-1=b) be such a,b-path. Select {u} = D \ P. Let P’= (x=a0, a=a1, a2,…, ak-1=b , ak=y), then P’ is a dominating set. Let sub-paths of P’: P1=(x=a0, a=a1, a2,…, ak-1=b) or P2=(a=a1, a2,…, ak-1=b , ak=y) is dominating set, it is Path-MCDS and completes the proof. Otherwise, a vertex x’ that xx’E, N[x’]P2= and a vertex y’ that yy’E, N[y’]P1=. We know ux’ and uy’  E, and thus (x, x’, u, y’, y) is a path whose length = 4 < 5.

17. Open problems • 1. Complete the proof for the results of path-MCDS of dominating pair graphs ! (the part of diameter=3) • 2. Find a polynomial time recognition algorithm for dominating pair graphs. • 3. Strong perfect graph conjecture is true for dominating pair graphs ? • A graph is perfect (minimum color number=maximum clique number) if and only if it contains no C2k+1 or the complement of C2k+1, for k  2.

18. n  1 1 2 3 n+1 A chordal graph is a dominating pair graph if and only if it doesn’t contain the following two graphs as induced subgraph. A chordal graph is a dominating pair graph if and only if no connected induced subgraph contains a 3-distant AT. A tree is a dominating pair graph if and only if it doesn’t contain the above left graph as induced subgraph.

19. n  1 n  1 1 2 n 1 2 3 n+1 A chordal graph is a AT-free graph if and only if it doesn’t contain the following four graphs as induced subgraph.