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ABSTRACT: [This journal has Romeo color yellow. So this is a preprint version of the paper.]
In previous work, two axiomatic characterizations were given for
the median function on median graphs: one involving the three simple and
natural axioms anonymity, betweenness and consistency;
the other involving faithfulness, consistency and $\frac{1}{2}$Condorcet.
To date, the independence of these axioms has not been a serious point of study. The aim of this paper is to provide the missing answers. The independent subsets of these five axioms are determined precisely and examples provided in each case on arbitrary median graphs. There are three cases that stand out. Here nontrivial examples and proofs are needed to give a full answer. Extensive use of the structure of median graphs is used throughout. Discrete Mathematics Algorithms and Applications 04/2015; DOI:10.1142/S1793830915500135

Source Available from: Henry Martyn Mulder
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ABSTRACT: In 1952 Sholander formulated an axiomatic characterization of the interval function of a tree with a partial proof. In 2011 Chvátal et al. gave a completion of this proof. In this paper we present a characterization of the interval function of a block graph using axioms on an arbitrary transit function RR. From this we deduce two new characterizations of the interval function of a tree. Discrete Mathematics 02/2015; 338:885  894. · 0.56 Impact Factor

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Clusters, Orders, and Trees: Methods and Applications, Edited by F. Aleskerov, B. Goldengorin, P.M. Pardalos, 05/2014: chapter 4: pages 63  75; Springer Verlag., ISBN: 9781493907427

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ABSTRACT: 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 studied. Here one maximizes the average distance to the clients. In this paper the mixed case is studied. Clients are represented by a profile, which is a sequence of vertices with repetitions allowed. In a signed profile each element is provided with a sign from {+,}. Thus one can take into account whether the client prefers the facility (with a + sign) or rejects it (with a  sign). The graphs for which all median sets, or all antimedian sets, are connected are characterized. Various consensus strategies for signed profiles are studied, amongst which Majority, Plurality and Scarcity. Hypercubes are the only graphs on which Majority produces the median set for all signed profiles. Finally, the antimedian sets are found by the Scarcity Strategy on e.g. Hamming graphs, Johnson graphs and halfcubes. Ars Mathematica Contemporanea 06/2013; 6:127  145. · 0.74 Impact Factor

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ABSTRACT: 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. Discrete Mathematics 02/2013; 313:2013. DOI:10.1016/j.disc.2012.09.022 · 0.56 Impact Factor

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ABSTRACT: A pathneighborhood graph is a connected graph in which every neighborhood induces a path. In the main results the 3sunfree pathneighborhood graphs are characterized. The 3sun is obtained from a 6cycle by adding three chords between the three pairs of vertices at distance 2. A P k graph is a pathneighborhood graph in which every neighborhood is a P k , where P k is the path on k vertices. The P k graphs are characterized for k≤4. Discussiones Mathematicae Graph Theory 01/2013; 4(4). DOI:10.7151/dmgt.1700 · 0.28 Impact Factor

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AKCE International Journal of Graphs and Combinatorics 01/2013; 10(1).

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ABSTRACT: A profile π=(x1,…,xk)π=(x1,…,xk), of length kk, in a finite connected graph GG is a sequence of vertices of GG, with repetitions allowed. A median xx of ππ is a vertex for which the sum of the distances from xx to the vertices in the profile is minimum. The median function finds the set of all medians of a profile. Medians are important in location theory and consensus theory. A median graph is a graph for which every profile of length 3 has a unique median. Median graphs have been well studied, possess a beautiful structure and arise in many arenas, including ternary algebras, ordered sets and discrete distributed lattices. They have found many applications, for instance in location theory, consensus theory and mathematical biology. Trees and hypercubes are key examples of median graphs.We establish a succinct axiomatic characterization of the median procedure on median graphs, settling a question posed implicitly by McMorris, Mulder and Roberts in 1998 [19]. We show that the median procedure can be characterized on the class of all median graphs with only three simple and intuitively appealing axioms, namely anonymity, betweenness and consistency. Our axiomatization is tight in the sense that each of these three axioms is necessary. We also extend a key result of the same paper, characterizing the median function for profiles of even length on median graphs. Discrete Applied Mathematics 01/2013; 161:838  846. DOI:10.1016/j.dam.2012.10.027 · 0.80 Impact Factor

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ABSTRACT: 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 is a typical location function for finding a location for an obnoxious facility. The `converse' of the antimedian function is the median function, where the distance sum is minimized. The median function is well studied. For instance it has been characterized axiomatically by three simple axioms on median graphs. The median function behaves nicely on many classes of graphs. In contrast the antimedian function does not have a nice behavior on most classes. So a nice axiomatic characterization may not be expected. In this paper such a characterization is obtained for the two classes of graphs on which the antimedian iswellbehaved: paths and hypercubes. Discrete Mathematics Algorithms and Applications 12/2012; 04(04). DOI:10.1142/S1793830912500541

Source Available from: Henry Martyn Mulder
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ABSTRACT: The general problem in location theory deals with functions that find sites to minimize some cost, or maximize some benefit, to a given set of clients. In the discrete case sites and clients are represented by vertices of a graph, in the continuous case by points of a network. The axiomatic approach seeks to uniquely distinguish certain specific location functions among all the arbitrary functions that address this problem by using a list of intuitively pleasing axioms. The median function minimizes the sum of the distances to the client locations. This function satisfies three simple and natural axioms: anonymity, betweenness, and consistency. They suffice on tree networks (continuous case) as shown by Vohra (1996) [19], and on cubefree median graphs (discrete case) as shown by McMorris et al. (1998) [9]. In the latter paper, in the case of arbitrary median graphs, a fourth axiom was added to characterize the median function. In this note we show that the above three natural axioms still suffice for the hypercubes, a special instance of arbitrary median graphs. Discrete Applied Mathematics 06/2011; 159(9):939944. DOI:10.1016/j.dam.2011.02.001 · 0.80 Impact Factor

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ABSTRACT: A pvalue of a sequence π = (x1, x2,…, xk) of elements of a finite metric space (X, d) is an element x for which is minimum. The function ℓp with domain the set of all finite sequences defined by ℓp(π) = {x: x is a pvalue of π} is called the ℓpfunction on X. The ℓpfunctions with p = 1 and p = 2 are the wellstudied median and mean functions respectively. In this article, the ℓpfunction on finite trees is characterized axiomatically. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012 Networks 01/2011; 60(2). DOI:10.1002/net.20463 · 0.83 Impact Factor

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Advances in interdisciplinary applied discrete mathematics, Edited by Henry Martyn Mulder, Hemanshu Kaul, 01/2011: pages 71  91; World Scientific Publishing Co., ISBN: 9789814299145

Source Available from: Henry Martyn Mulder
Advances in interdisciplinary applied discrete mathematics, Edited by Heney Martyn Mulder, Hemanshu Kaul, 01/2011: pages 225  241; World Scientific Publishing Co., ISBN: 9789814299145

Source Available from: Henry Martyn Mulder
Henry Martyn Mulder
Advances in interdisciplinary applied discrete mathematics, Edited by Henry Martyn Mulder, Hemanshu Kaul, 01/2011; World Scientific Publishing Co., ISBN: ISBN: 9789814299145

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ABSTRACT: This book is available at the Publisher: World Scientific Publishing Co, or via boostores or online bookstores. 01/2011; World Scientific Publishing Co., ISBN: ISBN: 9789814299145

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ABSTRACT: Maximal outerplanar graphs are characterized using three different classes of graphs. A pathneighborhood 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 edgetriangular 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 pathneighborhood graphs, the edgetriangular graphs having a tree as triangle graph. 01/2011; 52.

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Discrete Mathematics Algorithms and Applications 09/2010; 2(3):313  329. DOI:10.1142/S1793830910000681

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ABSTRACT: A. J. Goldman [Optimal center location in simple networks, Transportation Sci. 5, 212–221 (1971)] proved the classical result on how to find the medians for a set of clients in a tree using majority rule. Here the clients are located at vertices of the tree, and a median is a vertex in the tree that minimizes the sum of the distances to the locations of the clients. The majority rule can be rephrased as the majority strategy: if we are at vertex v, then we move to neighbor w of v if a majority of the clients is closer to w than to v. This strategy can be applied in any connected graph. In [H. M. Mulder, Discrete Appl. Math. 80, No. 1, 97–105 (1997; Zbl 0888.05025)] the question was answered for which connected graphs the majority strategy always produces the set of medians for any given set of clients: these are precisely the median graphs. This class of graphs has been wellstudied in the literature. In this paper we relax the majority strategy: instead of requiring a majority of the clients to be closer to w than to v, to move to w if there are more vertices closer to w than to v (thus ignoring the clients at equal distance from v and w). The main result of the paper is that the plurality strategy always produces the median set for any given set of clients if and only if all median sets are connected. We prove a similar result for the hill climbing strategy and for the steepest ascent hill climbing strategy. Australasian Journal of Combinatorics 01/2010; 46:191  202.

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ABSTRACT: The triangle graph T (G) of a graph G has the triangles of G as its vertices, and two vertices of T (G) are adjacent whenever as triangles in G they share an edge. Some basic facts for triangle graphs are deduced. An equivalence relation is introduced such that graphs in the same equivalence class of this relation have isomorphic triangle graphs. The graphs for which the triangle graph is a tree are characterized, and the graphs that are isomorphic to their triangle graph are characterized. Note that the latter result was obtained independently from the one existing in the literature [4]. For this reason this technical report is not submitted to any journal.

Source Available from: Henry Martyn Mulder
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ABSTRACT: The triangle graph T (G) of a graph G has the triangles of G as its vertices, and two vertices of T (G) are adjacent whenever as triangles in G they share an edge. Some basic facts for triangle graphs are deduced. An equivalence relation is introduced such that graphs in the same equivalence class of this relation have isomorphic triangle graphs. The graphs for which the triangle graph is a tree are characterized, and the graphs that are isomorphic to their triangle graph are characterized. Note that the latter result was obtained independently from the one existing in the literature [4]. For this reason this technical report is not submitted to any journal.