On cd-Coloring of $$\{P_5,K_4\}$$-free Chordal Graphs

Abstract

AbstractA k-cd-coloring of a graph G is a partition of the vertex set of G into k independent sets \(V_1,\ldots ,V_k\), where each \(V_i\) is dominated by some vertex of G. The least integer k such that G admits a k-cd-coloring is called the cd-chromatic number, \(\chi _{cd}(G)\), of G. We say that \(S \subseteq V(G)\) is a subclique in G if \(d_G(x,y)\ne 2\) for every \(x,y \in S\). The cardinality of a maximum subclique in G is called the subclique number, \(\omega _s(G)\), of G. Given a graph G and \(k\in \mathbb {N}\), the problem cd-Colorability checks whether \(\chi _{cd}(G)\le k\). The problem cd-Colorability is NP-complete for \(K_4\)-free graphs [Merounane et al., 2014], \(P_5\)-free graphs, and chordal graphs [Shalu et al., 2020]. In this paper, we show that the problem cd-Colorability is \(O(n^2)\)-time solvable in the intersection of the above graph classes (\(\{P_5,K_4\}\)-free chordal graphs). The problem Subclique takes a graph G and \(k \in \mathbb {N}\) as inputs and checks whether \(\omega _s(G)\ge k\). The Subclique problem is NP-complete for \(P_6\)-free graphs and bipartite gaphs [Shalu et al., 2017]. We prove that the problem Subclique is \(O(n^3)\)-time solvable in the class of \(P_6\)-free chordal bipartite graphs (a subclass of \(P_6\)-free bipartite graphs). In addition, we show that the cd-chromatic number and the subclique number are equal in these two graph classes.