Document Type

Article

Publication Date

6-4-2018

Department

Department of Mathematical Sciences

Abstract

A path in an edge-colored graph G is rainbow if no two edges of it are colored the same. The graph G is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph G is strongly rainbow-connected. The minimum number of colors needed to make G rainbow-connected is known as the rainbow connection number of G, and is denoted by src(G). Similarly, the minimum number of colors needed to make G strongly rainbow-connected is known as the strong rainbow connection number of G, and is denoted by src(G). We prove that for every k≥3, deciding whether src(G)≤k is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an n-vertex split graph with a factor of n1/2−ϵ for any ϵ>0 unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.

Publisher's Statement

© 2018 by the author(s). Article deposited here in compliance with publisher policies. Publisher's version of record: https://doi.org/10.23638/DMTCS-20-1-22

Publication Title

Discrete Mathematics & Theoretical Computer Science

Version

Publisher's PDF

Included in

Mathematics Commons

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