Subclasses of k-trees: Characterization and recognition

Authors:
L. Markenzon;C. M. Justel;N. Paciornik
Affiliations:
Núcleo de Computação Eletrônica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil;Instituto Militar de Engenharia, Rio de Janeiro, Brazil;Ministério da Ciência e Tecnologia, Brasília, Brazil
Venue:
Discrete Applied Mathematics - Special issue: IV ALIO/EURO workshop on applied combinatorial optimization
Year:
2006

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Abstract

A k-tree is either a complete graph on k vertices or a graph G=(V,E) that contains a vertex whose neighbourhood in G induces a complete graph on k vertices and whose removal results in a k-tree. We present two new subclasses of k-trees and their properties. First, we present the definition and characterization of k-path graphs, based on the concept of k-paths, that generalizes the classic concept of paths. We also introduce the simple-clique k-trees, of which the maximal outerplanar graphs and the planar 3-trees are particular cases. Based on Characterization Theorems, we show recognition algorithms for both families. Finally, we establish the inclusion relations among these new classes and k-trees.