What Is between Chordal and Weakly Chordal Graphs?

  • Authors:

  • Elad Cohen;Martin Charles Golumbic;Marina Lipshteyn;Michal Stern

  • Affiliations:

  • Caesarea Rothschild Institute, University of Haifa, Israel;Caesarea Rothschild Institute, University of Haifa, Israel;Caesarea Rothschild Institute, University of Haifa, Israel;Caesarea Rothschild Institute, University of Haifa, Israel and The Academic College of Tel-Aviv - Yaffo, Israel

  • Venue:
  • Graph-Theoretic Concepts in Computer Science
  • Year:
  • 2008

Quantified Score

Hi-index 0.00

Visualization

Abstract

An (h ,s ,t )-representation of a graph G consists of a collection of subtrees {S v | v *** V (G )} of a tree T , such that (i) the maximum degree of T is at most h , (ii) every subtree has maximum degree at most s , and (iii) there is an edge between two vertices in the graph if and only if the corresponding subtrees in T have at least t vertices in common. For example, chordal graphs correspond to [ *** , *** ,1] = [3,3,1] = [3,3,2] graphs (notation of *** here means that no restriction is imposed). We investigate the complete bipartite graph K 2,n and prove new theorems characterizing those K 2,n graphs that have an (h ,s ,2)-representation and those that have an (h ,s ,3)-representation. We characterize [3,2,4] graphs as equivalent to the 4-flower-free [2,4,4] graphs and give a recognition algorithm for [2,3,4] graphs. Based on these characterizations, we present new results that confirm that weakly chordal graphs, as opposed to chordal graphs, can not be characterized within the [h ,s ,t ] framework. Furthermore, we show a hierarchy of families of graphs between chordal and weakly chordal within the [h ,s ,t ] framework.