H.M. Mulder (Henry)
http://repub.eur.nl/ppl/3922/
List of Publicationsenhttp://repub.eur.nl/eur_logo.png
http://repub.eur.nl/
RePub, Erasmus University RepositoryAxiomatic characterization of the interval function of a block
graph
http://repub.eur.nl/pub/51745/
Wed, 20 Aug 2014 00:00:01 GMT<div>K. Balakrishnan</div><div>M. Changat</div><div>A.K. Lakshmikuttyamma</div><div>J. Mathews</div><div>H.M. Mulder</div>
__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 $R$. From this we deduce two new characterizations of the interval function of a tree.Five axioms for location functions on median graphs
http://repub.eur.nl/pub/51344/
Thu, 15 May 2014 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>B. Novick</div><div>R.C. Powers</div>
__Abstract__
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 ½-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.Path-neighborhood graphs
http://repub.eur.nl/pub/74902/
Wed, 16 Oct 2013 00:00:01 GMT<div>R.C. Laskar</div><div>H.M. Mulder</div>
An axiomatic approach to location functions on finite metric spaces
http://repub.eur.nl/pub/41360/
Mon, 09 Sep 2013 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>B. Novick</div><div>R.C. Powers</div>
A location function on a finite metric space (X, d) is a function on the set, X*, of all finite sequences of elements of X, to 2X\θ, 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. A tight axiomatization of the median procedure on median graphs
http://repub.eur.nl/pub/63998/
Tue, 01 Jan 2013 00:00:01 GMT<div>H.M. Mulder</div><div>B. Novick</div>
A profile π=(x1,.,xk), of length k, in a finite connected graph G is a sequence of vertices of G, with repetitions allowed. A median x of π is a vertex for which the sum of the distances from x 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.Guides and shortcuts in graphs
http://repub.eur.nl/pub/72723/
Tue, 01 Jan 2013 00:00:01 GMT<div>H.M. Mulder</div><div>L. Nebeský</div>
The geodesic structure of a graph 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 connected 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.Quorum Colorings of Graphs
http://repub.eur.nl/pub/37620/
Thu, 13 Sep 2012 00:00:01 GMT<div>S.M. Heditniemi</div><div>R.C. Laskar</div><div>H.M. Mulder</div>
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 $v$ 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.The ℓ
p-function on trees
http://repub.eur.nl/pub/34915/
Sat, 01 Sep 2012 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>O. Ortega</div>
A p-value of a sequence π = (x1, x2,⋯, xk) of elements of a finite metric space (X, d) is an element x for which ∑i=1kdp(x,xi) is minimum. The function ℓpwith domain the set of all finite sequences defined by ℓp(π) = {x: x is a p-value of π} is called the ℓp-function on X. The ℓp-functions with p = 1 and p = 2 are the well-studied median and mean functions respectively. In this article, the ℓp-function on finite trees is characterized axiomatically. Consensus strategies for signed profiles on graphs
http://repub.eur.nl/pub/37740/
Fri, 15 Jun 2012 00:00:01 GMT<div>K. Balakrishnan</div><div>M. Changat</div><div>H.M. Mulder</div><div>A.R. Subhamathi</div>
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.Consensus strategies for signed profiles on graphs
http://repub.eur.nl/pub/37742/
Fri, 15 Jun 2012 00:00:01 GMT<div>K. Balakrishnan</div><div>M. Changat</div><div>H.M. Mulder</div><div>A.R. Subhamathi</div>
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.Consensus Strategies for Signed Profiles on Graphs
http://repub.eur.nl/pub/26664/
Mon, 17 Oct 2011 00:00:01 GMT<div>K. Balakrishnan</div><div>M. Changat</div><div>H.M. Mulder</div><div>A.R. Subhamathi</div>
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.A simple axiomatization of the median procedure on median graphs
http://repub.eur.nl/pub/25628/
Mon, 01 Aug 2011 00:00:01 GMT<div>H.M. Mulder</div><div>B. Novick</div>
A profile = (x1, ..., xk), of length k, in a finite connected graph G is a sequence
of vertices of G, with repetitions allowed. A median x of is a vertex for which
the sum of the distances from x 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 are well studied. They
arise in many arenas, and have many applications.
We establish a succinct axiomatic characterization of the median procedure on
median graphs. This is a simplification of the characterization given by McMorris,
Mulder and Roberts [17] in 1998. We show that the median procedure can be characterized
on the class of all median graphs with only three simple and intuitively
appealing axioms: anonymity, betweenness and consistency. We also extend a key
result of the same paper, characterizing the median function for profiles of even
length on median graphs.
Maximal outerplanar graphs as chordal graphs, path-neighborhood graphs, and triangle graphs
http://repub.eur.nl/pub/23560/
Tue, 07 Jun 2011 00:00:01 GMT<div>R.C. Laskar</div><div>H.M. Mulder</div><div>B. Novick</div>
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.An axiomatization of the median procedure on the n-cube
http://repub.eur.nl/pub/25515/
Mon, 06 Jun 2011 00:00:01 GMT<div>H.M. Mulder</div><div>B. Novick</div>
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. Axiomatic Characterization of the Antimedian Function on Paths and Hypercubes
http://repub.eur.nl/pub/22803/
Fri, 04 Mar 2011 00:00:01 GMT<div>K. Balakrishnan</div><div>M. Changat</div><div>H.M. Mulder</div><div>A.R. Subhamathi</div>
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 is
well-behaved: paths and hypercubes.Guides and Shortcuts in Graphs
http://repub.eur.nl/pub/30589/
Sat, 01 Jan 2011 00:00:01 GMT<div>H.M. Mulder</div><div>L. Nebesky</div>
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.The Lp-function on trees
http://repub.eur.nl/pub/20773/
Thu, 23 Sep 2010 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>O. Ortega</div>
Axiomatic characterization of the mean function on trees
http://repub.eur.nl/pub/37668/
Wed, 01 Sep 2010 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>O. Ortega</div>
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.
An sxiomatization of the median procedure on the n-cube
http://repub.eur.nl/pub/20144/
Wed, 28 Jul 2010 00:00:01 GMT<div>H.M. Mulder</div><div>B. Novick</div>
The general problem in location theory deals with functions that find sites on a graph (discrete case) or network (continuous case) in such a way as to minimize some cost (or maximize some benefit) to a given set of clients represented by vertices on the graph or points on the network. The axiomatic approach seeks to uniquely distinguish, by using a list of intuitively pleasing axioms, certain specific location functions among all the arbitrary functions that address this problem. For the median function, which minimizes the sum of the distances to the client locations, three simple and natural axioms, anonymity, betweenness, and consistency suffice on tree networks (continuous case) as shown by Vohra, and on cube-free median graphs (discrete
case) as shown by McMorris et.al.. 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, at least for the hypercubes, a special instance of arbitrary median graphs, the above three natural axioms still suffice.The induced path function, monotonicity and betweenness
http://repub.eur.nl/pub/18216/
Sat, 06 Mar 2010 00:00:01 GMT<div>M. Changat</div><div>J. Mathew</div><div>H.M. Mulder</div>
The geodesic interval function I of a connected graph allows an axiomatic characterization involving axioms on the function only, without any reference to distance, as was shown by Nebeský [20]. Surprisingly, Nebeský [23] showed that, if no further restrictions are imposed, the induced path function J of a connected graph G does not allow such an axiomatic characterization. Here J(u,v) consists of the set of vertices lying on the induced paths between u and v. This function is a special instance of a transit function. In this paper we address the question what kind of restrictions could be imposed to obtain axiomatic characterizations of J. The function J satisfies betweenness if wset membership, variantJ(u,v), with w≠u, implies unegated set membershipJ(w,v) and xset membership, variantJ(u,v) implies J(u,x)subset of or equal toJ(u,v). It is monotone if x,yset membership, variantJ(u,v) implies J(x,y)subset of or equal toJ(u,v). In the case where we restrict ourselves to functions J that satisfy betweenness, or monotonicity, we are able to provide such axiomatic characterizations of J by transit axioms only. The graphs involved can all be characterized by forbidden subgraphs.