Showing papers on "K-tree published in 2006"

Emergence of large cliques in random scale-free networks

Abstract: In a network cliques are fully connected subgraphs that reveal which are the tight communities present in it. Cliques of size c > 3 are present in random Erdos and Renyi graphs only in the limit of diverging average connectivity. Starting from the finding that real scale-free graphs have large cliques, we study the clique number in uncorrelated scale-free networks finding both upper and lower bounds. Interestingly, we find that in scale-free networks large cliques appear also when the average degree is finite, i.e. even for networks with power law degree distribution exponents γ(2,3). Moreover, as long as γ < 3, scale-free networks have a maximal clique which diverges with the system size.

Maximal Cliques in Unit Disk Graphs: Polynomial Approximation

Abstract: We consider the problem of generating all maximal cliques in an unit disk graph. General algorithms to find all maximal cliques are exponential, so we rely on a polynomial approximation. Our algorithm makes use of certain key geographic structures of these graphs. For each edge, we limit the set of vertices that may form cliques with this as the longest edge. We then consider several characteristic shapes determined by that edge, and prove that all cliques having this as the longest edge, are included in one of the sets of vertices contained in these shapes. Our algorithm works in O(m¢ 2 ) time and generates O(m¢) cliques, where m is the number of edges in the graph and ¢ is its maximum degree. We also provide a modified version of the algorithm which improves the performance in many cases, albeit without affecting the worst case running time.

Improved exponential-time algorithms for treewidth and minimum fill-in

Abstract: Exact exponential-time algorithms for NP-hard problems is an emerging field, and an increasing number of new results are being added continuously. Two important NP-hard problems that have been studied for decades are the treewidth and the minimum fill problems. Recently, an exact algorithm was presented by Fomin, Kratsch, and Todinca to solve both of these problems in time O*(1.9601). Their algorithm uses the notion of potential maximal cliques, and is able to list these in time O*(1.9601 n ), which gives the running time for the above mentioned problems. We show that the number of potential maximal cliques for an arbitrary graph G on n vertices is O*(1.8135 n ), and that all potential maximal cliques can be listed in O*(1.8899 n ) time. As a consequence of this results, treewidth and minimum fill-in can be computed in O*(1.8899 n ) time.

Almost all webs are not rank-perfect

Abstract: Graphs with circular symmetry, called webs, are crucial for describing the stable set polytopes of two larger graph classes, quasi-line graphs[8,12] and claw-free graphs [7,8]. Providing a complete linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. Ben Rebea conjectured a description for quasi-line graphs, see [12]; Chudnovsky and Seymour [2] verified this conjecture recently for quasi-line graphs not belonging to the subclass of fuzzy circular interval graphs and showed that rank facets are required in this case only. Fuzzy circular interval graphs contain all webs and even the problem of finding all facets of their stable set polytopes is open. So far, it is only known that stable set polytopes of webs with clique number = 4 having non-rank facets [10_12,15]. In this paper we prove, building on a construction for non-rank facets from [16], that the stable set polytopes of almost all webs with clique number >= 5 admit non-rank facets. This adds support to the belief that these graphs are indeed the core of Ben Rebea's conjecture. Finally, we present a conjecture how to construct all facets of the stable set polytopes of webs

Computing Automorphism Groups of Chordal Graphs Whose Simplicial Components Are of Small Size

Abstract: It is known that any chordal graph can be uniquely decomposed into simplicial components. Based on this fact, it is shown that for a given chordal graph, its automorphism group can be computed in O((c! · n)O(1)) time, where c denotes the maximum size of simplicial components and n denotes the number of nodes. It is also shown that isomorphism of those chordal graphs can be decided within the same time bound. From the viewpoint of polynomial-time computability, our result strictly strengthens the previous ones respecting the clique number.

Graph relations, clique divergence and surface triangulations

Abstract: This work has two aims: first, we introduce a powerful technique for proving clique divergence when the graph satisfies a certain symmetry condition. Second, we prove that each closed surface admits a clique divergent triangulation. By definition, a graph is clique divergent if the orders of its iterated clique graphs tend to infinity, and the clique graph of a graph is the intersection graph of its maximal complete subgraphs. © 2005 Wiley Periodicals, Inc. J Graph Theory

On the maximal cliques in c-max-tolerance graphs and their application in clustering molecular sequences.

Abstract: Given a set S of n locally aligned sequences, it is a needed prerequisite to partition it into groups of very similar sequences to facilitate subsequent computations, such as the generation of a phylogenetic tree. This article introduces a new method of clustering which partitions S into subsets such that the overlap of each pair of sequences within a subset is at least a given percentage c of the lengths of the two sequences. We show that this problem can be reduced to finding all maximal cliques in a special kind of max-tolerance graph which we call a c-max-tolerance graph. Previously we have shown that finding all maximal cliques in general max-tolerance graphs can be done efficiently in O(n3 + out). Here, using a new kind of sweep-line algorithm, we show that the restriction to c-max-tolerance graphs yields a better runtime of O(n2 log n + out). Furthermore, we present another algorithm which is much easier to implement, and though theoretically slower than the first one, is still running in polynomial time. We then experimentally analyze the number and structure of all maximal cliques in a c-max-tolerance graph, depending on the chosen c-value. We apply our simple algorithm to artificial and biological data and we show that this implementation is much faster than the well-known application Cliquer. By introducing a new heuristic that uses the set of all maximal cliques to partition S, we finally show that the computed partition gives a reasonable clustering for biological data sets.

Boundary cliques, clique trees and perfect sequences of maximal cliques of a chordal graph

Abstract: We characterize clique trees of a chordal graph in their relation to simplicial vertices and perfect sequences of maximal cliques. We investigate boundary cliques defined by Shibata[23] and clarify their relation to endpoints of clique trees. Next we define a symmetric binary relation between the set of clique trees and the set of perfect sequences of maximal cliques. We describe the relation as a bipartite graph and prove that the bipartite graph is always connected. Lastly we consider to characterize chordal graphs from the aspect of non-uniqueness of clique trees.

Vertex partitions of chordal graphs

Abstract: A k-tree is a chordal graph with no (k + 2)-clique. An 𝓁-tree-partition of a graph G is a vertex partition of G into 'bags,' such that contracting each bag to a single vertex gives an 𝓁-tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ 𝓁 ≥0, every k-tree has an 𝓁-tree-partition in which each bag induces a connected ${\lfloor k} /(\ell+1) \rfloor$-tree. An analogous result is proved for oriented k-trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 167172, 2006 Supported by Government of Spain grant MEC SB2003-0270, and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224. Research initiated at the Department of Applied Mathematics and the Institute for Theoretical Computer Science at Charles University, Prague, Czech Republic. Supported by project LN00A056 of the Ministry of Education of the Czech Republic, and by the European Union Research Training Network COMBSTRU (Combinatorial Structure of Intractable Problems).

On non-rank facets of stable set polytopes of webs with clique number four

Abstract: Graphs with circular symmetry, called webs, are relevant for describing the stable set polytopes of two larger graph classes, quasi-line graphs and claw-free graphs. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number = 4 having non-rank facets. The aim of the present paper is to treat the remaining case with clique number =4: we provide an infinite sequence of such webs whose stable set polytopes admit non-rank facets. s of claw-free graphs is a long-standing problem [M. Grotschel, L. Lovasz, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, Berlin, 1988]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number = 4 having non-rank facets [J. Kind, Mobilitatsmodelle fur zellulare Mobilfunknetze: Produktformen und Blockierung, Ph.D. Thesis, RWTH Aachen, 2000; G. Oriolo, Clique family inequalities for the stable set polytope for quasi-line graphs, Discrete Appl. Math. 132 (2004) 185-201; T. Liebling, G. Oriolo, B. Spille, G. Stauffer, On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs, Math. Meth. Oper. Res. 59 (2004) 25-35]. The aim of the present paper is to treat the remaining case with clique number =4: we provide an infinite sequence of such webs whose stable set polytopes admit non-rank facets.

Generation of graphs with bounded branchwidth

Abstract: Branchwidth is a connectivity parameter of graphs closely related to treewidth Graphs of treewidth at most k can be generated algorithmically as the subgraphs of k-trees n this paper, we investigate the family of edge-maximal graphs of branchwidth k, that we call k-branches The k-branches are, just as the k-trees, a subclass of the chordal graphs where all minimal separators have size k However, a striking difference arises when considering subgraph-minimal members of the family Whereas Kk+1 is the only subgraph-minimal k-tree, we show that for any k ≥7 a minimal k-branch having q maximal cliques exists for any value of , except for k=8,q=2 We characterize subgraph-minimal k-branches for all values of k Our investigation leads to a generation algorithm, that adds one or two new maximal cliques in each step, producing exactly the k-branches

MATHEMATICAL ENGINEERING TECHNICAL REPORTS Boundary cliques, clique trees and perfect sequences of maximal cliques of a chordal graph

Abstract: We characterize clique trees of a chordal graph in their relation to simplicial vertices and perfect sequences of maximal cliques. We investigate boundary cliques defined by Shibata[23] and clarify their relation to endpoints of clique trees. Next we define a symmetric binary relation between the set of clique trees and the set of perfect sequences of maximal cliques. We describe the relation as a bipartite graph and prove that the bipartite graph is always connected. Lastly we consider to characterize chordal graphs from the aspect of non-uniqueness of clique trees.

Generation of graphs with bounded branchwidth

Abstract: Branchwidth is a connectivity parameter of graphs closely related to treewidth. Graphs of treewidth at most k can be generated algorithmically as the subgraphs of k-trees. n this paper, we investigate the family of edge-maximal graphs of branchwidth k, that we call k-branches. The k-branches are, just as the k-trees, a subclass of the chordal graphs where all minimal separators have size k. However, a striking difference arises when considering subgraph-minimal members of the family. Whereas K k+1 is the only subgraph-minimal k-tree, we show that for any k > 7 a minimal k-branch having q maximal cliques exists for any value of q ¬∈ {3, 5}, except for k = 8, q = 2. We characterize subgraph-minimal k-branches for all values of k. Our investigation leads to a generation algorithm, that adds one or two new maximal cliques in each step, producing exactly the k-branches.

Bounds On The Labelling Numbers Of Chordal Graphs.

Abstract: Motivated by the conjecture on the L(2, 1)-labelling number λ(G) of a graph G by Griggs and Yeh [2] and the question: “Is the upper bound (∆+3)/4 for λ(G) for chordal graphs with maximum degree ∆ is sharp?”, posed by Sakai [3], we study the bounds for λ(G) for chordal graphs in this paper. Let G be a chordal graph on n vertices with maximum degree ∆ and maximum clique number ω. We improve the upper bound (∆+3)/4 on λ(G) and the upper bound (∆ + 2d− 1)/4 on λd(G) with d ≥ 2, answering question of Sakai and improving results of Chang et al. Finally, we study the labelling numbers of r-power paths P r n on n vertices. We obtain λd(P r n) for small integers d ≥ 2 and r ≥ 2, and give a better bound of λd(P r n) for large integers d and r.