Roman domination on strongly chordal graphs

  • Authors:

  • Chun-Hung Liu;Gerard J. Chang

  • Affiliations:

  • Department of Mathematics, National Taiwan University, Taipei, Taiwan 10617 and School of Mathematics, Georgia Institute of Technology, Atlanta, USA 30332;Department of Mathematics, National Taiwan University, Taipei, Taiwan 10617 and Taida Institute for Mathematical Sciences, National Taiwan University, Taipei, Taiwan 10617 and National Center for ...

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2013

Quantified Score

Hi-index 0.00

Visualization

Abstract

Given real numbers b驴a0, an (a,b)-Roman dominating function of a graph G=(V,E) is a function f:V驴{0,a,b} such that every vertex v with f(v)=0 has a neighbor u with f(u)=b. An independent/connected/total (a,b)-Roman dominating function is an (a,b)-Roman dominating function f such that {v驴V:f(v)驴0} induces a subgraph without edges/that is connected/without isolated vertices. For a weight function $w{:} V\to\Bbb{R}$ , the weight of f is w(f)=驴 v驴V w(v)f(v). The weighted (a,b)-Roman domination number $\gamma^{(a,b)}_{R}(G,w)$ is the minimum weight of an (a,b)-Roman dominating function of G. Similarly, we can define the weighted independent (a,b)-Roman domination number $\gamma^{(a,b)}_{Ri}(G,w)$ . In this paper, we first prove that for any fixed (a,b) the (a,b)-Roman domination and the total/connected/independent (a,b)-Roman domination problems are NP-complete for bipartite graphs. We also show that for any fixed (a,b) the (a,b)-Roman domination and the total/connected/weighted independent (a,b)-Roman domination problems are NP-complete for chordal graphs. We then give linear-time algorithms for the weighted (a,b)-Roman domination problem with b驴a0, and the weighted independent (a,b)-Roman domination problem with 2a驴b驴a0 on strongly chordal graphs with a strong elimination ordering provided.