Showing papers on "K-tree published in 2013"
TL;DR: This paper improves the polynomial delay algorithm for sparse graphs and enumerates all maximal cliques in O(Δ⋅H3) time delay, where H is the so called H-value of a graph or equivalently it is the smallest integer satisfying |{v∈V∣δ(v)≥H}|≤H given δ( v) as the degree of a vertex.
Abstract: In this paper, we consider the problem of generating all maximal cliques in a sparse graph in polynomial delay. Given a graph G=(V,E) with n vertices and m edges, the latest and fastest polynomial delay algorithm for sparse graphs enumerates all maximal cliques in O(Δ
4) time delay, where Δ is the maximum degree of vertices. However, it requires an O(n⋅m) preprocessing time. We improve it in two aspects. First, our algorithm does not need preprocessing. Therefore, our algorithm is a truly polynomial delay algorithm. Second, our algorithm enumerates all maximal cliques in O(Δ⋅H
3) time delay, where H is the so called H-value of a graph or equivalently it is the smallest integer satisfying |{v∈V∣δ(v)≥H}|≤H given δ(v) as the degree of a vertex. In real-world network data, H usually is a small value and much smaller than Δ.
59 citations
TL;DR: A branch-and-bound algorithm for finding all cliques of size k in a k-partite graph is proposed that improves upon the method of Grunert et al.
Abstract: In this paper, a branch-and-bound algorithm for finding all cliques of size k in a k-partite graph is proposed that improves upon the method of Grunert et al. (in Comput Oper Res 29(1):13–31, 2002). The new algorithm uses bit-vectors, or bitsets, as the main data structure in bit-parallel operations. Bitsets enable a new form of data representation that improves branching and backtracking of the branch-and-bound procedure. Numerical studies on randomly generated instances of k-partite graphs demonstrate competitiveness of the developed method.
25 citations
TL;DR: This study produces a new family of Hansen polytopes that have only $3^d+16$ nonempty faces, and confirms Kalai's $3D$ conjecture for such poly topes and shows that the Hannerpolytopes among them correspond to threshold graphs.
Abstract: We analyze a remarkable class of centrally symmetric polytopes, the Hansen polytopes of split graphs. We confirm Kalai's $3^d$ conjecture for such polytopes (they all have at least $3^d$ nonempty faces) and show that the Hanner polytopes among them (which have exactly $3^d$ nonempty faces) correspond to threshold graphs. Our study produces a new family of Hansen polytopes that have only $3^d+16$ nonempty faces.
12 citations
TL;DR: In this paper, the connections between abstract polytopes and properly edge colored graphs are investigated, focussing in particular on aspects of symmetry and geometric, combinatorial, or algebraic properties.
Abstract: The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P
G
of rank n, the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic properties of the polytope P
G
and the combinatorial or algebraic structure of the underlying graph G, focussing in particular on aspects of symmetry. Several such families of colorful polytopes are studied including examples derived from a Cayley graph, in particular the graphicahedra, as well as the flagadjacency polytopes and related monodromy polytopes associated with a given abstract polytope. The duals of certain families of colorful polytopes have been important in the topological study of colored triangulations and crystallization of manifolds.
10 citations
Posted Content•
TL;DR: In this article, the maximum number of vertices of k-neighborly edge polytopes up to a sublinear term was determined for k = 2, 3, 5.
Abstract: The "edge polytope" of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k =2, 3, 5 we determine the maximum number of vertices of k-neighborly edge polytopes up to a sublinear term. We also construct a family of edge polytopes with exponentially-many facets.
9 citations
01 Jan 2013
TL;DR: The modified version of GJK algorithm for finding a common point of two polytopes is considered and knowing this point and using the duality theorem the intersection of twopolytopes can be found.
Abstract: The modified version of GJK algorithm for finding a common point of two polytopes is considered. Knowing this point and using the duality theorem the intersection of two polytopes can be found. Considering polytopes in pairs, we can find an intersection region of m polytopes.
6 citations
05 Jun 2013
TL;DR: This work considers one such problem, namely that of counting the number of maximal independent sets (MISs) in a graph, which is a well studied problem with applications in several research areas.
Abstract: There are a number of problems that require the counting or the enumeration of all occurrences of a certain structure within a given data set. We consider one such problem, namely that of counting the number of maximal independent sets (MISs) in a graph. Along with its complement problem of counting all maximal cliques, this is a well studied problem with applications in several research areas.
5 citations
TL;DR: The effect of operations like edge contraction, edge removal and others on the dynamical behavior of a graph under the iteration of the clique operator K is explored and it is proved that every clique Divergent graph is a spanning subgraph of a clique divergent graph with diameter two.
4 citations
19 Jun 2013
TL;DR: The clique arrangement \(\mathcal{A}\)(G) for a chordal graph G is presented to describe the intersections between the maximal cliques more precisely than in clique trees or related concepts.
Abstract: In this paper, we present the clique arrangement \(\mathcal{A}\)(G) for a chordal graph G to describe the intersections between the maximal cliques of G more precisely than in clique trees or related concepts. In particular, the node set of \(\mathcal{A}\)(G) consists of all intersections of maximal cliques of G. In \(\mathcal{A}\)(G), there is an arc from a node X to a node Z, if X is a subset of Z and there is no node Y, that is a superset of X and a subset of Z.
4 citations
TL;DR: A representation for chordal graphs called the compact representation, based on the running intersection property, is presented, which provides the means to immediately deduce several structural properties of a chordal graph such as a perfect elimination ordering, the minimal vertex separators and a clique-tree.
3 citations
TL;DR: The results seem to suggest that the topology still determines the K -behavior in these cases, and that there are no K -convergent triangulations of the sphere, the projective plane, the torus and the Klein bottle.
Posted Content•
TL;DR: In this article, a bijection between the set of clique trees and the product of local finite families of finite trees is introduced, and the edges of a clique tree are shown to be in bijection with edges of the corresponding collection of trees.
Abstract: We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting.
Journal Article•
TL;DR: In this article, it was shown that a geometric realization of the clique complex of a connected chordal graph is homologically trivial and as a consequence it is always the case for any connected chordial graph G that ∑_(k=1)^I ǫ(G)âÕ(k-1) I·_k (G)=1, where I Ã(G ) is the number of cliques of order k in G and Ãǫ (G) is the cliques number of G.
Abstract: It is shown that a geometric realization of the clique complex of a connected chordal graph is homologically trivial and as a consequence of this it is always the case for any connected chordal graph G that ∑_(k=1)^I‰(G)â–’(-1)^(k-1) I·_k (G)=1, where I·_k (G) is the number of cliques of order k in G and I‰(G) is the clique number of G.
01 Jan 2013
TL;DR: In this paper, it was shown that k-dominating is polynomial-time solvable for strongly chordal graphs, and that the problem is also solvable in counting monadic second-order logic.
Abstract: Due to its large range of applications, many variations and extensions of the classical domination problem in graphs have been defined and studied during the past fourty years. Given a graph G = (V,E), A ⊆ R and B = {b1, . . . , b|V |}, an A,B-dominating function of G is a function f : V 7→ A such that f(N [vi]) ≥ bi for all v ∈ V , where f(U) = ∑ u∈U f(u), for U ⊆ V and N [v] is the closed neighborhood of v. The weigth of f is given by w(f) = f(V ). This work is focused in two variations of the problem. Let k ∈ Z+ and bi = k for all i ∈ {1, . . . , |V |}. When A = {0, 1}, f is a k-tuple dominating function and γ×k(G) is the k-tuple domination number of G [3]. When A = {0, 1, . . . , k}, f is a {k}-dominating function and γ{k}(G) is the {k}-domination number of G [1]. As usual, these definitions induce the study of the following decision problems, for fixed k ∈ Z+: k-TUPLE DOMINATING FUNCTION (k-DOM) Instance: G = (V,E), j ∈ N Question: Does G have a k-tuple dominating function of weight at most j? {k}-DOMINATING FUNCTION ({k}-DOM) Instance: G = (V,E), j ∈ N Question: Does G have a {k}-dominating function of weight at most j? In this work we obtain a new graph class where {k}-DOM is NP-complete: the class of chordal graphs. We also identify some maximal subclasses for which it is polynomial time solvable. By relating this problem with k-DOM, we prove that {k}-DOM is polynomial time solvable for strongly chordal graphs. Besides, by expressing the property involved in k-DOM in Counting Monadic Secondorder Logic, we obtain that both problems are linear time solvable for bounded tree-width graphs. In this way we enlarge the family of graphs for which k-DOM is polynomial time solvable (see [2]).
TL;DR: In this paper, the number of disjoint cliques in the complement graph and the sum of permanent functions over all principal minors of the adjacency matrix of the graph is computed.
Abstract: We give a new recursive method to compute the number of cliques and cycles of a graph. This method is related, respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph. In particular, let $G$ be a graph and let $overline {G}$ be its complement, then given the chromatic polynomial of $overline {G}$, we give a recursive method to compute the number of cliques of $G$. Also given the adjacency matrix $A$ of $G$ we give a recursive method to compute the number of cycles by computing the sum of permanent function of the principal minors of $A$. In both cases we confront to a new computable parameter which is defined as the number of disjoint cliques in $G$.
TL;DR: An algorithm is presented, which takes grid clique as a whole to find its all instance points neighbor lists in 9-neighbor area, and inserts them into the candidate extension clique list, then deletes the records that dont meet the criteria in the list to generate maximal clique.
Abstract: In this paper, an algorithm is presented, which takes grid clique as a whole to find its all instance points neighbor lists in 9-neighbor area, and inserts them into the candidate extension clique list, then deletes the records that dont meet the criteria in the list, finally combines grid cliques and candidate extension clique list to generate maximal clique. The paper discusses the characteristics of spatial co-location pattern algorithm, and makes statistics of its maximal clique number generated in different distance threshold. The experiment shows that the improved algorithm can find out more maximal cliques.