Transit functions on graphs (and posets)

The notion of transit function is introduced to present a unifying approachfor results and ideas on intervals, convexities and betweenness in graphs andposets. Prime examples of such transit functions are the interval function I andthe induced path function J of a connected graph. Another transit function isthe all-paths function. New transit functions are introduced, such as the cutvertextransit function and the longest path function. The main idea of transitfunctions is that of ‘transferring’ problems and ideas of one transit functionto the other. For instance, a result on the interval function I might suggestsimilar problems for the induced path function J. Examples are given of howfruitful this transfer can be. A list of Prototype Problems and Questions forthis transferring process is given, which suggests many new questions and openproblems.graph theory;betweenness;block graph;convexity;distance in graphs;interval function;path function;induced path;paths and cycles;transit function;types of graphs

Axiomatic characterization of the absolute median on cube-free median networks

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 of roads found in cities equipped with the Manhattan metric.

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).

Median computation in graphs using consensus strategies

Following the Majority Strategy in graphs, other consensus strategies, namely Plurality Strategy, Hill Climbing and Steepest Ascent Hill Climbing strategies on graphs are discussed as methods for the computation of median sets of profiles. A review ofalgorithms for median computation on median graphs is discussed and their time complexities are compared. Implementation of the consensus strategies on median computation in arbitrary graphs is discussed.majority strategy;consensus strategy;Hill climbing median computation

Axiomization of the center function on trees.

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

Network analysis in the Caribbean

The Caribbean region is a cross road of international and regional containertraffic. Most of the islands in the region have also adopted ambitiousstrategies to become prime locations for container transshipment. This paperintroduces a tool that can be used to visualise and analyse the Caribbeancontainer flows. The tool is constructed on the basis of a multi-layeredgraph structure and is highly parameterized to allow for flexible selectionof flows and ports. The tool supports the assessment of the potential forthe development of logistics and transport hub through the calculation ofrelevant indicators using available information on container flows in theregion. Much of the empirical work centers on the estimation of theorigin-destination matrix of container flows in the region. The paperpresents a case study for the island of Curacao.Caribbean;container flows;hub port;network analysis

Generalized centrality in trees

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 and Slater's path analogues for D = 2. We generalize these results concerning paths and include proofs that each type of generalized center and generalized centroid is unique. We also present algorithms for finding one or all generalized central substructures of each type.

The induced path function, monotonicity and betweenness

The induced path function $J(u, v)$ of a graph consists of the set of all vertices lying on the induced paths between vertices $u$ and $v$. This function is a special instance of a transit function. The function $J$ satisfies betweenness if $w \\in J(u, v)$ implies $u \\notin J(w, v)$ and $x \\in J(u, v)$ implies $J(u, x \\subseteq J(u, v)$, and it is monotone if $x, y \\in J(u, v)$ implies $J(x, y) \\subseteq J(u, v)$. The induced path function of aconnected graph satisfying the betweenness and monotone axioms are characterized by transit axioms.betweenness;induced path;transit function;monotone;house domino;long cycle;p-graph

Convexities related to path properties on graphs; a unified approach

Path properties, such as 'geodesic', 'induced', 'all paths' define a convexity on a connected graph. The general notion of path property, introduced in this paper, gives rise to a comprehensive survey of results obtained by different authors for a variety of path properties, together with a number of new results. We pay special attention to convexities defined by path properties on graph products and the classical convexity invariants, such as the Caratheodory, Helly and Radon numbers in relation with graph invariants, such as clique numbers and other graph properties.

Maximal outerplanar graphs as chordal graphs, path-neighborhood graphs, and triangle graphs

Maximal outerplanar graphs are characterized using three different classes of graphs. A path-neighborhood graph is a connected graph in which every neighborhood induces a path. The triangle graph $T(G)$ has the triangles of the graph $G$ as its vertices, two of these being adjacent whenever as triangles in $G$ they share an edge. A graph is edge-triangular if every edge is in at least one triangle. The main results can be summarized as follows: the class of maximal outerplanar graphs is precisely the intersection of any of the two following classes: the chordal graphs, the path-neighborhood graphs, the edge-triangular graphs having a tree as triangle graph.maximal outerplanar graph;path-neighborhood graph;triangle graph;chordal graph;elimination ordering