The induced path function, monotonicity and betweenness

  • Authors:

  • Manoj Changat;Joseph Mathew;Henry Martyn Mulder

  • Affiliations:

  • Department of Futures Studies, University of Kerala, Trivandrum-695 034, India;Department of Mathematics, S.B. College, Changanassery-686 101, India;Econometrisch Instituut, Erasmus Universiteit, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

The geodesic interval function I of a connected graph allows an axiomatic characterization involving axioms on the function only, without any reference to distance, as was shown by Nebesky [20]. Surprisingly, Nebesky [23] showed that, if no further restrictions are imposed, the induced path function J of a connected graph G does not allow such an axiomatic characterization. Here J(u,v) consists of the set of vertices lying on the induced paths between u and v. This function is a special instance of a transit function. In this paper we address the question what kind of restrictions could be imposed to obtain axiomatic characterizations of J. The function J satisfies betweenness if w@?J(u,v), with wu, implies u@?J(w,v) and x@?J(u,v) implies J(u,x)@?J(u,v). It is monotone if x,y@?J(u,v) implies J(x,y)@?J(u,v). In the case where we restrict ourselves to functions J that satisfy betweenness, or monotonicity, we are able to provide such axiomatic characterizations of J by transit axioms only. The graphs involved can all be characterized by forbidden subgraphs.