Four classes of perfectly orderable graphs

Abstract

A graph is called "perfectly orderable" if its vertices can be ordered in such a way that, for each induced subgraph F, a certain "greedy" coloring heuristic delivers an optimal coloring of F. No polynomial-time algorithm to recognize these graphs is known. We present four classes of perfectly orderable graphs: Welsh-Powell perfect graphs, Matula perfect graphs, graphs of Dilworth number at most three, and unions of two threshold graphs. Graphs in each of the first three classes are recognizable in a polynomial time. In every graph that belongs to one of the first two classes, we can find a largest clique and an optimal coloring in a linear time.

Maximum weight stable set in (P7, bull)-free graphs and (S1,2Maffray1, Pastor2 2018Discrete Mathematics6120View full textAdd to dashboardCite

“…By results from [6] and [12], MWSS can be solved in time O(n 3 ) in graphs with no C 5 , P 5 and P 5 . Hence, since the A i 's are pairwise disjoint, MWSS can be solved in time…”

A note on vertex orders for stability number

“…We shall remark that orderings of vertices similar to Conditions 2 and 3 have been studied in the context of vertex colorings, for instance in [1,[6][7][8][9][10]. The following two theorems can be proved similar to Theorem 2.2.…”

Contraction Blockers for Graphs with Forbidden Induced Paths

“…We only have to swap Cases 2 and 3 after observing that χ(G x ) = max{χ(G y ), χ(G z )} if x is a ⊕-node with y and z as its two children and χ(G x ) = χ(G y ) + χ(G z ) if x is a ⊗-node. We can use the same arguments as used in the proof for n = α for the running time analysis as well; we only have to observe that it takes O(n + m) time to compute the chromatic number of a cograph (using the same arguments as before or another algorithm of [8]). …”