Showing papers on "K-tree published in 2004"

Finding All Maximal Cliques in Dynamic Graphs

Abstract: Clustering applications dealing with perception based or biased data lead to models with non-disjunct clusters. There, objects to be clustered are allowed to belong to several clusters at the same time which results in a fuzzy clustering. It can be shown that this is equivalent to searching all maximal cliques in dynamic graphs like Gt e (V,Et), where Et − 1 ⊂ Et, t e 1,…,T; E0 e p. In this article algorithms are provided to track all maximal cliques in a fully dynamic graph.

The Worst-Case Time Complexity for Generating All Maximal Cliques (Extended Abstract)

Abstract: We present a depth-first search algorithm for generating all maximal cliques of an undirected graph, in which pruning methods are employed as in Bron and Kerbosch's algorithm. All maximal cliques generated are output in a tree-like form. Then we prove that its worst-case time complexity is O(3 n/3 ) for an n-vertex graph. This is optimal as a function of n, since there exist up to 3 n/3 cliques in an n-vertex graph.

Antiwebs are rank-perfect

Abstract: Webs and antiwebs are natural generalizations of odd holes and odd antiholes with circular symmetry of their maximum cliques and stable sets. Webs and antiwebs turned out to play a crucial role for describing the stable set polytopes for larger graph classes. In this short note we obtain, with the help of a result of Shepherd (1995), a complete description of the stable set polytopes for antiwebs showing that antiwebs are rank-perfect.

Flag Vectors of Multiplicial Polytopes

Abstract: Bisztriczky introduced the multiplex as a generalization of the simplex. A polytope is multiplicial if all its faces are multiplexes. In this paper it is proved that the flag vectors of multiplicial polytopes depend only on their face vectors. A special class of multiplicial polytopes, also discovered by Bisztriczky, is comprised of the ordinary polytopes . These are a natural generalization of the cyclic polytopes. The flag vectors of ordinary polytopes are determined. This is used to give a surprisingly simple formula for the $h$-vector of the ordinary $d$-polytope with $n+1$ vertices and characteristic $k$: $h_i={k-d+i\choose i}+(n-k){k-d+i-1\choose i-1}$, for $i\le d/2$. In addition, a construction is given for 4-dimensional multiplicial polytopes having two-thirds of their vertices on a single facet, answering a question of Bisztriczky.

A Note on Cliques in Multipartite Graphs

Abstract: We consider a set of cliques in any multipartite graph with two vertices in each part. Moreover, we construct a class of peculiar polytopes. Key words: multipartite graph, clique, polytope.

A Symbolic Analysis Method for Network Function Sensitivity Based on Network Graph Theory

Abstract: The sensitivity analysis is important in electric network optimum design and tolerance analysis. The general method of sensitivity analysis is pure numerical calculation, their characteristics are completly numerical analysis one by one. Those methods inevitably exist computational complexity, such as redundant polynomials eliminating, needing more data memory capacity and calculation error. Based on network graph theory, using admittance product of k-tree branches in connected graph, the symbolic expression of network function and its sensitivity are presented. Comparing with incremental network method, the method is more fast and effective and is prone to program. It is proved by the example that the method has validity and feasibility.

On the arrangement of cliques in chordal graphs with respect to the cuts

Abstract: A cut (A, B) (where B = V - A) in a graph G = (V, E) is called internal if and only if there exists a vertex x in A that is not adjacent to any vertex in B and there exists a vertex y is an element of B such that it is not adjacent to any vertex in A. In this paper, we present a theorem regarding the arrangement of cliques in a chordal graph with respect to its internal cuts. Our main result is that given any internal cut (A, B) in a chordal graph G, there exists a clique with kappa(G) + vertices (where kappa(G) is the vertex connectivity of G) such that it is (approximately) bisected by the cut (A, B). In fact we give a stronger result: For any internal cut (A, B) of a chordal graph, and for each i, 0 <= i <= kappa(G) + 1 such that vertical bar K-i vertical bar = kappa(G) + 1, vertical bar A boolean AND K-i vertical bar = i and vertical bar B boolean AND K-i vertical bar = kappa(G) + 1 - i. An immediate corollary of the above result is that the number of edges in any internal cut (of a chordal graph) should be Omega(k(2)), where kappa(G) = k. Prompted by this observation, we investigate the size of internal cuts in terms of the vertex connectivity of the chordal graphs. As a corollary, we show that in chordal graphs, if the edge connectivity is strictly less than the minimum degree, then the size of the mincut is at least kappa(G)(kappa(G)+1)/2 where kappa(G) denotes the vertex connectivity. In contrast, in a general graph the size of the mincut can be equal to kappa(G). This result is tight.