Roman Domination on Strongly Chordal Graph 呂紹甲 資工碩一 9.15

Outline Roman domination Strongly chordal graph

Roman Domination A Roman dominating function on a graph G =(V, E) is a function f : V →{0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V )= Σu∈V f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. Roman Domination on general graphs is a NP-hard problem.

Roman domination For example 2 1

Roman domination For example 1 2 1 2 2

Odd chord ˙A chord XiXj in a cycle C = (X1,X2,…,X2K) of even length 2K is an odd chord if the distance in C between Xi and Xj is odd.

Odd chord For example 2K=6 u v z

Odd chord For example 2K=8 u w v z

Elimination ordering ˙ The ordering (v1,v2,…,vn) of the vertex set of a graph G is an elimination ordering satisfying property P if for all i∈{1,2,…,n}, the vertex vi has property P in the remaining graph Gi =G({vi,…,vn}). ˙For example - Kn

perfect elimination ordering The vertex v∈V is simplicial in G if N(v) is clique in G. The ordering (v1,v2,…,vn) of the vertices of V is a perfect elimination ordering of G if for all i∈ {1,2,…,n}, the vertex vi is simplicial in Gi = G({vi,….vn}).

perfect elimination ordering For example 1 6 5 2 3 4

strong perfect elimination ordering ˙A perfect elimination ordering of G is a strong perfect elimination ordering of G if the following condition is fulfilled : For each i < j< k < l , if vi is adjacent to vk and vl , and vj is adjacent to vk , then vj is adjacent to vl.

strong perfect elimination ordering For example 3<4<5<6 1 3 6 2 5 4

strongly perfect elimination ordering A vertex ordering (v1,v2,…,vn) of a graph G = (V,E) is a strongly perfect elimination ordering if and only if it fulfills the following the two conditions : 1 (v1,v2,…,vn) is a perfect elimination ordering, 2 For each i < j and k < l with vivl ∈ E, vivk ∈E , and vkvl it follows that vlvj ∈E.

strong perfect elimination ordering For example 1<5 and 2<3 2<6 and 3<5 3<4 and 5<6 3<5 and 2<6 1 3 6 2 5 4

Strongly Chordal Graph A graph is strongly chordal if and only if it admit a strongly perfect elimination ordering. G is strongly chordal if G is chordal and each cycle in G of even length at least 6 has an odd chord.

Strongly Chordal Graph cycle of length 6 u v

Strongly Chordal Graph cycle of length 8 u v