H.M. Mulder (Henry)
http://repub.eur.nl/ppl/3922/
List of Publicationsenhttp://repub.eur.nl/logo.jpg
http://repub.eur.nl/
RePub, Erasmus University RepositoryFive 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.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. 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.Axiomatic Characterization of the Mean Function on Trees
http://repub.eur.nl/pub/18261/
Tue, 23 Feb 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 paper the mean function on finite trees is characterized axiomatically.On the Economic Order Quantity Model With Transportation Costs
http://repub.eur.nl/pub/16675/
Wed, 09 Sep 2009 00:00:01 GMT<div>S.I. Birbil</div><div>K. Bulbul</div><div>J.B.G. Frenk</div><div>H.M. Mulder</div>
We consider an economic order quantity type model with unit out-of-pocket holding costs, unit
opportunity costs of holding, fixed ordering costs and general transportation costs. For these models, we analyze
the associated optimization problem and derive an easy procedure for determining a bounded interval containing
the optimal cycle length. Also for a special class of transportation functions, like the carload discount schedule, we
specialize these results and give fast and easy algorithms to calculate the optimal lot size and the corresponding
optimal order-up-to-level.Axiomatic characterization of the interval function of a graph
http://repub.eur.nl/pub/13768/
Mon, 10 Nov 2008 00:00:01 GMT<div>H.M. Mulder</div><div>L. Nebesky</div>
A fundamental notion in metric graph theory is that of the interval function I : V × V → 2V – {} of a (finite) connected graph G = (V,E), where I(u,v) = { w | d(u,w) + d(w,v) = d(u,v) } is the interval between u and v. An obvious question is whether I can be characterized in a nice way amongst all functions F : V × V 2V – {}. This was done in [13, 14, 16] by axioms in terms of properties of the functions F. The authors of the present paper, in the conviction that characterizing the interval function belongs to the central questions of metric graph theory, return here to this result again. In this characterization the set of axioms consists of five simple, and obviously necessary, axioms, already presented in [9], plus two more complicated axioms. The question arises whether the last two axioms are really necessary in the form given or whether simpler axioms would do the trick. This question turns out to be non-trivial. The aim of this paper is to show that these two supplementary axioms are optimal in the following sense. The functions satisfying only the five simple axioms are studied extensively. Then the obstructions are pinpointed why such functions may not be the interval function of some connected graph. It turns out that these obstructions occur precisely when either one of the supplementary axioms is not satisfied. It is also shown that each of these supplementary axioms is independent of the other six axioms. The presented way of proving the characterizing theorem (Theorem 3 here) allows us to find two new separate ``intermediate'' results (Theorems 1 and 2). In addition some new characterizations of modular and median graphs are presented. As shown in the last section the results of this paper could provide a new perspective on finite connected graphs.Entrepreneurship Education and Training in a Small Business Context: Insights from the Competence-based Approach
http://repub.eur.nl/pub/12466/
Thu, 22 May 2008 00:00:01 GMT<div>T. Lans</div><div>W. Hulsink</div><div>H. Baert</div><div>H.M. Mulder</div>
The concept of competence, as it is brought into play in current research, is a potentially powerful construct for entrepreneurship education research and practice. Although the concept has been the subject of strong debate in educational research in general, critical analysis of how it has been used, applied and experienced in entrepreneurship education practice is scarce. This article contributes specifically to the discussion of entrepreneurial competence by theoretically unfolding and discussing the concept. Subsequently, the implications of applying a competence-based approach in entrepreneurship education are illustrated and discussed based on analysis of two cases that were aimed at identifying, diagnosing and eventually developing entrepreneurial competence in small businesses in the Netherlands and Flanders (Belgium). The cases show that the added value of focussing on competence in entrepreneurship education lies in making the (potential) small business owner aware of the importance of certain entrepreneurial competencies and in providing direction for competence development. In this process it is fundamental that competence is treated as an item for discussion and interpretation, rather than as a fixed template of boxes to be ticked. Furthermore the cases highlight that a competence-based approach does not determine the type of educational and instructional strategies to be used. Its consequential power in that respect is limited.