## 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.

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The authors wish to thank the anonymous referees for their valuable comments on the content and presentation of this paper.

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Shalu, M.A., Vijayakumar, S., Yamini, S.D. *et al.* On the algorithmic aspects of strong subcoloring.
*J Comb Optim* **35**, 1312–1329 (2018). https://doi.org/10.1007/s10878-018-0272-z

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DOI: https://doi.org/10.1007/s10878-018-0272-z