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International Conference on Combinatorial Optimization and Applications

COCOA 2008: Combinatorial Optimization and Applications pp 160–170Cite as

Simplicial Powers of Graphs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5165))

Abstract

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 = (V G ,E G ) is the k-simplicial power of a graph H = (V H ,E H ) (H a root graph of G) if V G is the set of all simplicial vertices of H, and for all distinct vertices x and y in V G , xy ∈ E G 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 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.

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Authors and Affiliations

  1. Institut für Informatik, Universität Rostock, D-18051, Rostock, Germany

    Andreas Brandstädt & Van Bang Le

Authors
  1. Andreas Brandstädt
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  2. Van Bang Le
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Boting Yang Ding-Zhu Du Cao An Wang

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Brandstädt, A., Le, V.B. (2008). Simplicial Powers of Graphs. In: Yang, B., Du, DZ., Wang, C.A. (eds) Combinatorial Optimization and Applications. COCOA 2008. Lecture Notes in Computer Science, vol 5165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85097-7_15

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  • DOI: https://doi.org/10.1007/978-3-540-85097-7_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85096-0

  • Online ISBN: 978-3-540-85097-7

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