A Representation Diagram for Maximal Independent Sets of a Graph

Masakuni TAKI  Sumio MASUDA  Toshinobu KASHIWABARA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E81-A   No.5   pp.784-788
Publication Date: 1998/05/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
maximal independent set,  interval graph,  cocomparability graph,  generation problem,  

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Let H=(V(H),E(H)) be a directed graph with distinguished vertices s and t. An st-path in H is a simple directed path starting from s and ending at t. Let (H) be defined as { SS is the set of vertices on an st-path in H (s and t are excluded)}. For an undirected graph G=(V(G),E(G)) with V(G) V(H)- { s,t }, if the family of maximal independent sets of G coincides with (H), we call H an MIS-diagram for G. In this paper, we provide a necessary and sufficient condition for a directed graph to be an MIS-diagram for an undirected graph. We also show that an undirected graph G has an MIS-diagram iff G is a cocomparability graph. Based on the proof of the latter result, we can construct an efficient algorithm for generating all maximal independent sets of a cocomparability graph.