Recently in several papers, graphs with maximum neighborhood orderings were characterized and turned out to be algorithmically useful. This paper gives a unified framework for characterizations of those graphs in terms of neighborhood and clique hypergraphs which have the Helly property and whose line graph is chordal. These graphs are dual (in the sense of hypergraphs) to chordal graphs. By using the hypergraph approach in a systematical way new results are obtained, some of the old results are generalized, and some of the proofs are simplified.

Abstract not found

. We are interested in the problem of how an agent organizes its sensorimotor experiences in order to create a spatial representation. Our approach to solve this problem is the Spatial Semantic Hierarchy #SSH#, an ontological hierarchy of representations for knowledge of largescale space. At the SSH topological level, space is represented by places and connectivity relationships among them. Places are arranged into paths so that the topological representation looks like the street network of a city. Grouping places into regions allows an agent to reason e#ciently about its spatial knowledge. Regions can be organized in a hierarchical structure suitable for hierarchical planning and human-level interface. In this paper we showhow a hierarchy of regions can be automatically created by an agent. We extend the SSH axiomatic theory to include regions as #rst order objects at the SSH topological level. Based on this formalization, an implementation using Annotated Hierarchical ...

A graph is dually chordal if it is the clique graph of a chordal graph. Alternatively, a graph is dually chordal if it admits a maximum neighbourhood order. This class generalizes known subclasses of chordal graphs such as doubly chordal graphs, strongly chordal graphs and interval graphs. We prove that Vizing's total-colour conjecture holds for dually chordal graphs. We describe a new heuristic that yields an exact total-colouring algorithm for even maximum degree dually chordal graphs and an exact edge-colouring algorithm for odd maximum degree dually chordal graphs. Key words. graph algorithms, chordal graphs, total colouring, edge colouring, clique graphs, maximum neighbourhood ordering. AMS subject classification. 05C85, 05C15, 68R10, 90C27 1 Introduction We consider the problem of total colouring and edge colouring dually chordal graphs. A total colouring of a graph is a colouring of its vertices and edges such that no adjacent vertices, no adjacent edges, and no incident ver...

Abstract not found