On the algorithmic aspects of strong subcoloring

M. A. Shalu (), S. Vijayakumar (), S. Devi Yamini () and T. P. Sandhya ()
Additional contact information
M. A. Shalu: Indian Institute of Information Technology, Design and Manufacturing (IIITD&M)
S. Vijayakumar: Indian Institute of Information Technology, Design and Manufacturing (IIITD&M)
S. Devi Yamini: VIT University
T. P. Sandhya: The Hong Kong Polytechnic University

Journal of Combinatorial Optimization, 2018, vol. 35, issue 4, No 17, 1312-1329

Abstract: Abstract A partition of the vertex set V(G) of a graph G into $$V(G)=V_1\cup V_2\cup \cdots \cup V_k$$ V ( G ) = V 1 ∪ V 2 ∪ ⋯ ∪ V k is called a k-strong subcoloring if $$d(x,y)\ne 2$$ d ( x , y ) ≠ 2 in G for every $$x,y\in V_i$$ x , y ∈ V i , $$1\le i \le k$$ 1 ≤ i ≤ k where d(x, y) denotes the length of a shortest x-y path in G. The strong subchromatic number is defined as $$\chi _{sc}(G)=\text {min}\{ k:G \text { admits a }k$$ χ sc ( G ) = min { k : G admits a k - $$\text {strong subcoloring}\}$$ 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 $$K_{1,r}$$ K 1 , r -free split graphs, StrongSubcoloring is in P when $$r\le 3$$ r ≤ 3 and NP-complete when $$r>3$$ r > 3 and (ii) for the class of H-free graphs, StrongSubcoloring is polynomial time solvable only if H is an induced subgraph of $$P_4$$ 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 $$x,y\in S$$ x , y ∈ S , $$d(x,y)= 2$$ d ( x , y ) = 2 in G and the cardinality of a maximum strong set in G is denoted by $$\alpha _{s}(G)$$ α s ( G ) . Clearly, $$\alpha _{s}(G)\le \chi _{sc}(G)$$ α s ( G ) ≤ χ 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) $$K_{1,4}$$ K 1 , 4 -free split graphs, and it is polynomial time solvable for (i) trees and (ii) $$P_4$$ P 4 -free graphs.

Keywords: Coloring; Subcoloring; Strong subcoloring; Strong set (search for similar items in EconPapers)
Date: 2018
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10878-018-0272-z Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jcomop:v:35:y:2018:i:4:d:10.1007_s10878-018-0272-z

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10878

DOI: 10.1007/s10878-018-0272-z

Access Statistics for this article

Journal of Combinatorial Optimization is currently edited by Thai, My T.

More articles in Journal of Combinatorial Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().