Leaps: an approach to the block structure of a graph

To study the block structure of a connected graph G=(V,E), we introduce two algebraic approaches that reflect this structure: a binary operation + called a leap operation and a ternary relation L called a leap system, both on a finite, nonempty set V. These algebraic structures are easily studied by considering their underlying graphs, which turn out to be block graphs. Conversely, we define the operation +G as well as the set of leaps LG of the connected graph G. The underlying graph of +G , as well as that of LG , turns out to be just the block closure of G (i.e. the graph obtained by making each block of G into a complete subgraph).

Guides and Shortcuts in Graphs

The geodesic structure of a graphs appears to be a very rich structure. There are many ways to describe this structure, each of which captures only some aspects. Ternary algebras are for this purpose very useful and have a long tradition. We study two instances: signpost systems, and a special case of which, step systems. Signpost systems were already used to characterize graph classes. Here we use these for the study of the geodesic structure of a spanning subgraph F with respect to its host graph G. Such a signpost system is called a guide to (F,G). Our main results are: the characterization of the step system of a cycle, the characterization of guides for spanning trees and hamiltonian cycles

Axiomatic characterization of the interval function of a graph

A fundamental notion in metric graph theory is that of the interval function I : V × V → 2V – {∅} of a (finite) connected graph G = (V,E), where I(u,v) = { w | d(u,w) + d(w,v) = d(u,v) } is the interval between u and v. An obvious question is whether I can be characterized in a nice way amongst all functions F : V × V -> 2V – {∅}. This was done in [13, 14, 16] by axioms in terms of properties of the functions F. The authors of the present paper, in the conviction that characterizing the interval function belongs to the central questions of metric graph theory, return here to this result again. In this characterization the set of axioms consists of five simple, and obviously necessary, axioms, already presented in [9], plus two more complicated axioms. The question arises whether the last two axioms are really necessary in the form given or whether simpler axioms would do the trick. This question turns out to be non-trivial. The aim of this paper is to show that these two supplementary axioms are optimal in the following sense. The functions satisfying only the five simple axioms are studied extensively. Then the obstructions are pinpointed why such functions may not be the interval function of some connected graph. It turns out that these obstructions occur precisely when either one of the supplementary axioms is not satisfied. It is also shown that each of these supplementary axioms is independent of the other six axioms. The presented way of proving the characterizing theorem (Theorem 3 here) allows us to find two new separate ``intermediate'' results (Theorems 1 and 2). In addition some new characterizations of modular and median graphs are presented. As shown in the last section the results of this paper could provide a new perspective on finite connected graphs

Leaps: an approach to the block structure of a graph

To study the block structure of a connected graph G=(V,E), we introduce two algebraic approaches that reflect this structure: a binary operation + called a leap operation and a ternary relation L called a leap system, both on a finite, nonempty set V. These algebraic structures are easily studied by considering their underlying graphs, which turn out to be block graphs. Conversely, we define the operation +G as well as the set of leaps LG of the connected graph G. The underlying graph of +G , as well as that of LG , turns out to be just the block closure of G (i.e. the graph obtained by making each block of G into a complete subgraph)