Henry Martyn Mulder

Distance-hereditary graphs

Abstract Distance-hereditary graphs (sensu Howorka) are connected graphs in which all induced paths are isometric. Examples of such graphs are provided by complete multipartite graphs and ptolemaic graphs. Every finite distance-hereditary graph is obtained from K 1 by iterating the following two operations: adding pendant vertices and splitting vertices. Moreover, distance-hereditary graphs are characterized in terms of the distance function d , or via forbidden isometric subgraphs.

The median procedure on median graphs

Abstract A median of a profile π = (x1, …, xk) of vertices of a finite connected graph G is a vertex x for which ∑ki = 1 d(x, xi) is minimum, where d is the usual geodesic distance on G. The function Med whose domain is the set of all profiles and is given by Med(π) = {x: x is a median of π} is called the median procedure on G. In this paper, the median procedure is characterized for median graphs and cube-free median graphs.

Quasi-median graphs and algebras

Scattered through the literature various structures occur that generalize median graphs and median algebras: quasi-median graphs, retracts of Hamming graphs, graphs of finite windex, isotropic media, subdirect products of simplex algebras, certain ternary algebras, and so on. We connect them all up, provide some new generalizations of quasi-median graphs, some new sets of axioms for quasi-median algebras, and some shorter proofs as well.

The all-paths transit function of a graph

AbstractA transit function R on a set V is a function

Median Graphs and Triangle-Free Graphs

R:VxV \to 2^2

The median function on median graphs and semilattices

satisfying the axioms

Tolerance intersection graphs on binary trees with constant tolerance

u \in R(u,\upsilon ),R(u,\upsilon ) = R(\upsilon ,u)