Graphs on make domination

Chordal and near chordal graphs contain rich structure features and these structure features can lead to efficient solutions for some computationally intractable problems. We recall the following theorems. Precomputed dominating sets for many named graphs can be obtained in the Wolfram Language using GraphData [ graph , "DominationNumber" ]. This polynomial has been introduced by the author in his Ph. Linear FPT reductions and computational lower bounds. Now, we show that G has a dominating set of size k if and only if H contains an independent dominating set of size k. It is interesting that for the classification of graphs with exactly two, three, and four domination roots, we must consider some kinds of corona of two graphs.

We show that the Set Cover problem can be parametrically reduced to this problem in polynomial time. For example, the Independent Set problem is to decide whether a graph contains an independent set of size k or not and has been shown to be W[1]-hard [ 9 ]. Practice online or make a printable study sheet. Let be a simple graph. Based on the same arguments in the proof of Theorem 1, the minimum dominating set in graph H contains at most OPT I vertices. In this paper, we study the parameterized complexity of Dominating Set problem in chordal graphs and near chordal graphs. The intersection graph of all created subtrees is a chordal graph H.

A root of is called a domination root of. Every root of is called the domination root of. In this paper, we introduce the concept of k -power domination which is a common generalization of domination and power domination. Walk through homework problems step-by-step from beginning to end. Figure 1 a provides an illustration of the tree T constructed from the set S.

The domination number should not be confused with the domatic number , which is the maximum size of a domatic partition in a graph. These solutions are often guaranteed to be within a ratio of the optimal solution and thus can be useful in practice. In [ 14 ], it is conjectured that an odd-chordal graph is perfect if its complement is also odd-chordal. Complexity and Lower Bound. Journal of Combinatorial Theory, Series B. You have access to this content.