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On clique relaxation models in network analysis
 European Journal of Operational Research
, 2013
"... Increasing interest in studying community structures, or clusters in complex networks arising in various applications has led to a large and diverse body of literature introducing numerous graphtheoretic models relaxing certain characteristics of the classical clique concept. This paper analyzes th ..."
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Increasing interest in studying community structures, or clusters in complex networks arising in various applications has led to a large and diverse body of literature introducing numerous graphtheoretic models relaxing certain characteristics of the classical clique concept. This paper analyzes the elementary cliquedefining properties implicitly exploited in the available clique relaxation models and proposes a taxonomic framework that not only allows to classify the existing models in a systematic fashion, but also yields new clique relaxations of potential practical interest. Some basic structural properties of several of the considered models are identified that may facilitate the choice of methods for solving the corresponding optimization problems. In addition, bounds describing the cohesiveness properties of different clique relaxation structures are established, and practical implications of choosing one model over another are discussed.
The Complete Catalog of 3Regular, Diameter3 Planar Graphs
, 1996
"... The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 ..."
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The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 vertices. An Upper Bound A graph with maximum degree \Delta and diameter D is called a (\Delta; D)graph. It is easily seen ([9], p. 171) that the order of a (\Delta,D)graph is bounded above by the Moore bound, which is given by 1+ \Delta + \Delta (\Delta \Gamma 1) + \Delta \Delta \Delta + \Delta(\Delta \Gamma 1) D\Gamma1 = 8 ? ! ? : \Delta(\Delta \Gamma 1) D \Gamma 2 \Delta \Gamma 2 if \Delta 6= 2; 2D + 1 if \Delta = 2: Figure 1: The regular (3,3)graph on 20 vertices (it is unique up to isomorphism) . For D 2 and \Delta 3, this bound is attained only if D = 2 and \Delta = 3; 7, and (perhaps) 57 [3, 14, 23]. Now, except for the case of C 4 (the cycle on four vertices), the num...
Distance Connectivity in Graphs and Digraphs ∗
"... Let G = (V, A) be a digraph with diameter D � = 1. For a given integer 2 ≤ t ≤ D, the tdistance connectivity κ(t) of G is the minimum cardinality of an x → y separating set over all the pairs of vertices x, y which are at distance d(x, y) ≥ t. The tdistance edge connectivity λ(t) of G is defined ..."
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Let G = (V, A) be a digraph with diameter D � = 1. For a given integer 2 ≤ t ≤ D, the tdistance connectivity κ(t) of G is the minimum cardinality of an x → y separating set over all the pairs of vertices x, y which are at distance d(x, y) ≥ t. The tdistance edge connectivity λ(t) of G is defined similarly. The tdegree of G, δ(t), is the minimum among the outdegrees and indegrees of all vertices with (out or in)eccentricity at least t. A digraph is said to be maximally distance connected if κ(t) = δ(t) for all values of t. In this paper we give a construction of a digraph having D − 1 positive arbitrary integers c2 ≤... ≤ cD, D> 3, as the values of its tdistance connectivities κ(2) = c2,..., κ(D) = cD. Besides, a digraph that shows the independence of the parameters κ(t), λ(t), and δ(t) is constructed. Also we derive some results on the distance connectivities of digraphs, as well as sufficient conditions for a digraph to be maximally distance connected. Similar results for (undirected) graphs are presented. 1