On the algorithmic aspects of strong subcoloring

Abstract

A partition of the vertex set V(G) of a graph G into $V(G)=V_1\cup V_2\cup \cdots \cup V_k$ is called a k-strong subcoloring if $d(x,y)\ne 2$ in G for every $x,y\in V_i$, $1\le i \le k$ where d(x, y) denotes the length of a shortest x-y path in G. The strong subchromatic number is defined as $\chi _{sc}(G)=\text {min}\{ k:G \text { admits a }k$-$\text {strong subcoloring}\}$. In this paper, we explore the complexity status of the StrongSubcoloring problem: for a given graph G and a positive integer k, StrongSubcoloring is to decide whether G admits a k-strong subcoloring. We prove that StrongSubcoloring is NP-complete for subcubic bipartite graphs and the problem is polynomial time solvable for trees. In addition, we prove the following dichotomy results: (i) for the class of $K_{1,r}$-free split graphs, StrongSubcoloring is in P when $r\le 3$ and NP-complete when $r>3$ and (ii) for the class of H-free graphs, StrongSubcoloring is polynomial time solvable only if H is an induced subgraph of $P_4$; otherwise the problem is NP-complete. Next, we consider a lower bound on the strong subchromatic number. A strong set is a set S of vertices of a graph G such that for every $x,y\in S$, $d(x,y)= 2$ in G and the cardinality of a maximum strong set in G is denoted by $\alpha _{s}(G)$. Clearly, $\alpha _{s}(G)\le \chi _{sc}(G)$. We consider the complexity status of the StrongSet problem: given a graph G and a positive integer k, StrongSet asks whether G contains a strong set of cardinality k. We prove that StrongSet is NP-complete for (i) bipartite graphs and (ii) $K_{1,4}$-free split graphs, and it is polynomial time solvable for (i) trees and (ii) $P_4$-free graphs.