Discussiones Mathematicae Graph Theory 27(2) (2007) 345-357
doi: 10.7151/dmgt.1366

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HISTORIES IN PATH GRAPHS

Ludovít Niepel

Department of Mathematics and Computer Science
Kuwait University
P.O. Box 5969, Safat, 13060, Kuwait
e-mail: niepel@mcs.sci.kuniv.edu.kw

Abstract

For a given graph G and a positive integer r the r-path graph, Pr(G), has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length r−1, and their union forms either a cycle or a path of length k+1 in G. Let Pkr(G) be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of Pkr(G). The k-history P−kr(H) is a subgraph of G that is induced by all edges that take part in the recursive definition of H. We present some general properties of k-histories and give a complete characterization of graphs that are k-histories of vertices of 2-path graph operator.

Keywords: path-graph, graph dynamics, history.

2000 Mathematics Subject Classification: 05C38, 05C75.

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Received 15 May 2006
Revised 15 January 2007
Accepted 15 January 2007