Simplicial Powers of Graphs

  • Authors:

  • Andreas Brandstädt;Van Bang Le

  • Affiliations:

  • Institut für Informatik, Universität Rostock, Rostock, Germany D-18051;Institut für Informatik, Universität Rostock, Rostock, Germany D-18051

  • Venue:
  • COCOA 2008 Proceedings of the 2nd international conference on Combinatorial Optimization and Applications
  • Year:
  • 2008

Quantified Score

Hi-index 0.00

Visualization

Abstract

In a finite simple undirected graph, a vertex is simplicial if its neighborhood is a clique. We say that, for k驴 2, a graph G= (VG,EG) is the k-simplicial powerof a graph H= (VH,EH) (Ha root graphof G) if VGis the set of all simplicial vertices of H, and for all distinct vertices xand yin VG, xy驴 EGif and only if the distance in Hbetween xand yis at most k. This concept generalizes k-leaf powers introduced by Nishimura, Ragde and Thilikos which were motivated by the search for underlying phylogenetic trees; k-leaf powers are the k-simplicial powers of trees. Recently, a lot of work has been done on k-leaf powers and their roots as well as on their variants phylogenetic roots and Steiner roots. For k驴 {3,4,5}, k-leaf powers can be recognized in linear time, and for k驴 {3,4}, structural characterizations are known. For all other k, recognition and structural characterization of k-leaf powers is open.Since trees and block graphs (i.e., connected graphs whose blocks are cliques) have very similar metric properties, it is natural to study k-simplicial powers of block graphs. We show that leaf powers of trees and simplicial powers of block graphs are closely related, and we study simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs.