Simplicial Powers of Graphs

In a finite simple undirected graph, a vertex is simplicial if its neighborhood is a clique. We say that, for k 2, a graph G = (VG, EG) is the k-simplicial power of a graph H = (VH, EH) (H a root graph of G) if VG is the set of all simplicial vertices of H, and for all distinct vertices x and y in VG, xy EG if and only if the distance in H between x and y is 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 {3, 4, 5}, k-leaf powers can be recognized in linear time, and for k {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 c...

• Type: Conference
• Year: 2008
• Where: COCOA
• Authors: Andreas Brandstädt, Van Bang Le
• Keywords: COCOA 2008, Discrete Geometry, K-leaf Powers, Simple Undirected Graph, Simplicial