Publications

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    F.R. McMorris · Henry Martyn Mulder · Beth Novick · Robert C. Powers
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    ABSTRACT: This is a preprint. It will appear in Discrete Applied Mathematics. It is already online. A location problem can often be phrased as a consensus problem. The median function is a location/consensus function on a connected graph that has the finite sequences of vertices of as input. For each such sequence , returns the set of vertices that minimize the distance sum to the elements of . The median function satisfies three intuitively clear axioms: Anonymity, Betweenness and Consistency. Mulder and Novick showed in 2013 that on median graphs these three axioms actually characterize . This result raises a number of questions:
    Full-text · Article · Jan 2016
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    F. R. Mcmorris · Henry Martyn Mulder · Beth Novick · R. C. Powers
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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 non-trivial examples and proofs are needed to give a full answer. Extensive use of the structure of median graphs is used throughout.
    Full-text · Article · Apr 2015 · Discrete Mathematics Algorithms and Applications
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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.
    Full-text · Article · Feb 2015 · Discrete Mathematics
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    F.R. McMorris · Henry Martyn Mulder · Fred S. Roberts

    Full-text · Chapter · May 2014
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    F.R. McMorris · H.M. Mulder · B. Novick · R.C. Powers
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    ABSTRACT: A location function on a finite metric space ( ) is a function on the set, , of all finite sequences of elements of X, to , which minimizes some criteria of remoteness. Axiomatic characterizations of these functions have, for the most part, been established only for very special cases. While McMorris, Mulder and Powers [F.R. McMorris, H.M. Mulder, R.C. Powers, “The median function on median graphs and semilattices,” Discrete Appl. Math., 101, (2000), 221–230] were able to characterize the median function on median graphs with three axioms, one of their axioms was very specific to the structure of median graphs. Recently, however, Mulder and Novick [H.M. Mulder, B.A. Novick, “A tight axiomatization of the median procedure on median graphs,” Discrete Appl. Math., 161, (2013), 838–846] characterized the median function for all median graphs using only three very natural axioms. These three axioms are meaningful in the more general context of finite metric spaces. In this work, we establish that these same three axioms are indeed independent and then we settle completely the question of interdependence among the collection of axioms involved in the above mentioned two characterizations, giving examples for all logically relevant cases. We introduce several new location functions and pose some questions.
    Full-text · Article · Sep 2013 · Electronic Notes in Discrete Mathematics
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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.
    Full-text · Article · Jun 2013 · Ars Mathematica Contemporanea
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    Henry Martyn Mulder · Beth Novick
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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.
    Full-text · Article · Apr 2013 · Discrete Applied Mathematics
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    Henry Martyn Mulder · Ladislav Nebesky
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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.
    Full-text · Article · Feb 2013 · Discrete Mathematics
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    R.C. Laskar · Henry Martyn Mulder
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    ABSTRACT: A path-neighborhood graph is a connected graph in which every neighborhood induces a path. In the main results the 3-sun-free path-neighborhood graphs are characterized. The 3-sun is obtained from a 6-cycle by adding three chords between the three pairs of vertices at distance 2. A P k -graph is a path-neighborhood 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.
    Full-text · Article · Jan 2013 · Discussiones Mathematicae Graph Theory
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    ABSTRACT: Let G = (V, E) be a graph. A partition π = {V1, V2,., Vk} of the vertex set V of G into k color classes Vi, with 1 ≤ i ≤ k, is called a quorum coloring if for every vertex v ∈ V, at least half of the vertices in the closed neighborhood N[v] of v have the same color as v. In this paper we introduce the study of quorum colorings of graphs and show that they are closely related to the concept of defensive alliances in graphs. Moreover, we determine the maximum quorum coloring of a hypercube.
    Full-text · Article · Jan 2013 · AKCE International Journal of Graphs and Combinatorics
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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 iswell-behaved: paths and hypercubes.
    Full-text · Article · Dec 2012 · Discrete Mathematics Algorithms and Applications
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    Henry Martyn Mulder · Beth Novick
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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 cube-free 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.
    Full-text · Article · Jun 2011 · Discrete Applied Mathematics
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    F. R. McMorris · Henry Martyn Mulder · O. Ortega
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    ABSTRACT: A p-value of a sequence pi=(x(1), x(2),..., x(k)) of elements of a finite metric space (X, d) is an element x for which Sigma(k)(i=1) d(p)(x, x(1)) is minimum. The function l(p) with domain the set of all finite sequences defined by l(p)(pi) = {x: x is a p-value of pi} is called the l(p)-function on X. The l(p)-functions with p = 1 and p = 2 are the well-studied median and mean functions respectively. In this article, the l(p)-function on finite trees is characterized axiomatically. (C) 2011 Wiley Periodicals, Inc.
    Full-text · Article · Jan 2011 · Networks
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    Henry Martyn Mulder · F.R. McMorris · Rakesh V. Vohra

    Full-text · Chapter · Jan 2011
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    Full-text · Chapter · Jan 2011
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    Henry Martyn Mulder

    Full-text · Chapter · Jan 2011
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    Henry Martyn Mulder · Hemanshu Kaul
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    ABSTRACT: This book is available at the Publisher: World Scientific Publishing Co, or via boostores or online bookstores.
    Full-text · Book · Jan 2011
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    R. C. Laskar · H. M. Mulder · Novick B. B
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    ABSTRACT: 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.
    Full-text · Article · Jan 2011
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    F.R. McMorris · Henry Martyn Mulder · Oscar Ortega
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    ABSTRACT: A mean 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 Mean whose domain is the set of all finite sequences on X and is defined by Mean (π) = {x|x is a mean of π} is called the mean function on X. In this note, the mean function on finite trees is characterized axiomatically.
    Full-text · Article · Sep 2010 · Discrete Mathematics Algorithms and Applications
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    Henry Martyn Mulder · Kannan Balakrishnan · M. Changat
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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 well-studied 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.
    Full-text · Article · Jan 2010 · Australasian Journal of Combinatorics

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