Algorithmic aspect of stratified domination in graphs

• Published in: Information processing letters Vol. 113; no. 22-24; pp. 861 - 865
• Main Authors: Jennhwa Chang, Gerard, Chang, Chan-Wei, Kuo, David, Poon, Sheung-Hung
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.11.2013; Elsevier Sequoia S.A
• Subjects: Algorithms; Bipartite graph; Chordal graph; Coloring; Computer science; Data processing; Design of algorithms; Domination; F-domination; Graph algorithms; Graph coloring; Graphs; k-stratified graph; Mathematical problems; NP-complete; Planar graph; Reproduction; Studies; Tree; Trees; Domination; k-stratified graph; Design of algorithms; F-domination; Planar graph; Chordal graph; Tree; NP-complete; Graph algorithms; Bipartite graph
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2013.08.008

Chartrand, Haynes, Henning and Zhang introduced a variation of domination called stratified domination in graphs. This paper studies stratified domination from an algorithmic point of view. A 2-stratified (or black–white) graph is a graph in which every vertex is colored black or white. Given a black-white graph F rooted at a white vertex v, an F-coloring of a graph G=(V,E) is a black-white coloring of V for which every white vertex v of G belongs to a copy of F (not necessarily induced in G) rooted at v. An F-dominating set of G is the set of all black vertices in an F-coloring. The F-domination number γF(G) of G is the minimum cardinality of an F-dominating set. We consider the 3-vertex black-white graph F3 rooted at a white vertex v adjacent to another white vertex u, which adjacent to a black vertex w. We prove that the F3-domination problem is NP-complete for bipartite planar graphs and for chordal graphs. We also give a linear-time algorithm for the F3-domination problem in trees. •The F3-domination problem is NP-complete for bipartite planar graphs.•The F3-domination problem is NP-complete for chordal graphs.•A linear-time algorithm for the F3-domination problem in trees.