Powers of distance-hereditary graphs

Powers of distance-hereditary graphs need not be distance-hereditary, but they come close : the house, the domino, and cycles of length larger than four are forbidden induced subgraphs, whereas the 5-vertex fan may occur. Moreover, all even powers are chordal, and an odd power is chordal if and only if the given distance-hereditary graph has no induced subgraph of a certain type (viz., a 4-cycle with pendant paths of suitable length). In addition, one can exhibit further small graphs that are forbidden (as induced subgraphs) in either all or just some powers of distance-hereditary graphs. Other invariants of these powers can be expressed in terms of distances (yielding a 4-point condition) or intersection patterns (viz., Helly type conditions) of disksSIGLEAvailable from TIB Hannover: RO 1945(220) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Powers of distance-hereditary graphs

Powers of distance-hereditary graphs need not be distance-hereditary, but they come close : the house, the domino, and cycles of length larger than four are forbidden induced subgraphs, whereas the 5-vertex fan may occur. Moreover, all even powers are chordal, and an odd power is chordal if and only if the given distance-hereditary graph has no induced subgraph of a certain type (viz., a 4-cycle with pendant paths of suitable length). In addition, one can exhibit further small graphs that are forbidden (as induced subgraphs) in either all or just some powers of distance-hereditary graphs. Other invariants of these powers can be expressed in terms of distances (yielding a 4-point condition) or intersection patterns (viz., Helly type conditions) of disksSIGLEAvailable from TIB Hannover: RO 1945(220) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Powers of HHD-free graphs

The k-th power of a graph G is the graph with the same vertex set as G where two vertices are adjacent if their distance in G is at most k. In this paper we consider HHD-free graphs, i.e. the graphs where each cycle of length at least five has two chords. We show, that odd powers of HHD-free graphs are again HHD-free, and characterize those HHD-free graphs by forbidden subgraphs for which odd powers are chordal, and even powers are HHD-free or chordal respectively. (orig.)Available from TIB Hannover: RO 1945(315) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Convexity and HHD-free graphs

It is well-known that chordal graphs can be characterized via m-convexity. In this paper we introduce the notion of m"3-convexity (a relaxation of m-convexity) which is closely related to semisimplicial orderings of graphs. Using this notion we give simpler proofs of results from [8]and present some new characterizations of HHD-free graphs via m"3-convexity of disks. Moreover, we characterize weak bipolarizable graphs as the graphs for which the family of all m"3-convex sets is a convex geometry. As an application of our results we present a simple efficient criterion for deciding whether a HHD-free graph with given vertex radius function r is r-dominated by a clique. (orig.)Available from TIB Hannover: RR 1945(290) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman