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...

In 1982, Slater defined path subgraph analogues to the center, median, and (branch or branchweight) centroid of a tree. We define three families of central substructures of trees, including three types of central subtrees of degree at most D that yield the center, median, and centroid for D = 0...

Let $G = (V,E)$ be a graph. A partition $\pi = \{V_1, V_2, \ldots, V_k \}$ of the vertices $V$ of $G$ into $k$ {\it color classes} $V_i$, with $1 \leq i \leq k$, is called a {\it quorum coloring} if for every vertex $v \in V$, at least half of the vertices in the closed neighborhood $N[v]$ of...

The Majority Strategy for finding medians of a set of clients on a graph can be relaxed in the following way: if we are at v, then we move to a neighbor w if there are at least as many clients closer to w than to v (thus ignoring the clients at equal distance from v and w). The graphs on which...

The notion of transit function is introduced to present a unifying approach for results and ideas on intervals, convexities and betweenness in graphs and posets. Prime examples of such transit functions are the interval function I and the induced path function J of a connected graph. Another...

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...

In Vohra, European J. Operational Research 90 (1996) 78 – 84, a characterization of the absolute median of a tree network using three simple axioms is presented. This note extends that result from tree networks to cube-free median networks. A special case of such networks is the grid structure...

We give a new, short proof that four certain axiomatic properties uniquely define the center of a tree.

The median problem is a classical problem in Location Theory: one searches for a location that minimizes the average distance to the sites of the clients. This is for desired facilities as a distribution center for a set of warehouses. More recently, for obnoxious facilities, the antimedian was...

An antimedian of a profile $\\pi = (x_1, x_2, \\ldots , x_k)$ of vertices of a graph $G$ is a vertex maximizing the sum of the distances to the elements of the profile. The antimedian function is defined on the set of all profiles on $G$ and has as output the set of antimedians of a profile. It...