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

2. Outline
Roman domination
Strongly chordal graph

3. 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.

4. Roman domination
For example 2 1

5. Roman domination
For example 1 2 1 2 2

6. 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.

7. Odd chord
For example
2K=6 u v z

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

9. 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

10. 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}).

11. perfect elimination ordering
For example 1 6 5 2 3 4

12. 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.

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

14. 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.

15. 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

16. 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.

17. Strongly Chordal Graph
cycle of length 6 u v

18. Strongly Chordal Graph
cycle of length 8 u v