Showing papers on "K-tree published in 1999"
TL;DR: It is proved that, in the new formulation, there is a one-to-one correspondence between maximal cliques and maximal subtree isomorphisms, which allows the tree matching problem to be cast as an indefinite quadratic program using the Motzkin-Straus theorem.
Abstract: It is well-known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a (derived) "association graph." However, it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized, i.e., are trees, since maximal cliques are not constrained to preserve the partial order. We provide a solution to the problem of matching two trees by constructing the association graph using the graph-theoretic concept of connectivity. We prove that, in the new formulation, there is a one-to-one correspondence between maximal cliques and maximal subtree isomorphisms. This allows us to cast the tree matching problem as an indefinite quadratic program using the Motzkin-Straus theorem, and we use "replicator" dynamical systems developed in theoretical biology to solve it. Such continuous solutions to discrete problems are attractive because they can motivate analog and biological implementations. The framework is also extended to the matching of attributed trees by using weighted association graphs. We illustrate the power of the approach by matching articulated and deformed shapes described by shock trees.
435 citations
TL;DR: In this article, it was shown that for each chordal graph, there is a tree such that for every two vertices, the distance between them inTis not larger than the distance inGplus 2.
93 citations
TL;DR: In this article, the authors provided examples of k-divergent graphs such that the diameters of the iterated clique graphs also tend to infinity with n and the sizes of the cliques and even the chromatic numbers remain bounded.
25 citations
TL;DR: The strength of an edge or cycle is the number of maximal complete subgraphs it is in, and among hereditary clique-Helly graphs, this is equivalent to the subgraph of edges of strength at least k always forming a chordal graph.
8 citations
17 Jun 1999
TL;DR: The proposed technique is used to find approximated solutions for a class of related optimization problems where a solution corresponds to an acyclic orientation of graphs.
Abstract: We study the problem of finding an acyclic orientation of an undirected graph G such that each path is contained in a limited number of maximal cliques of G. In general, in an acyclic oriented graph, each path is contained in more than one maximal cliques. We focus our attention on crown-free interval graphs, and show how to find an acyclic orientation of such a graph, which guarantees that each path is contained in at most four maximal cliques. The proposed technique is used to find approximated solutions for a class of related optimization problems where a solution corresponds to an acyclic orientation of graphs.
2 citations
TL;DR: In this article, a particular partition of the vertex set of claw-free strongly chordal graphs in which each element is a clique was found, and it was shown that the adjacency graph of these cliques is a tree.
Abstract: In this paper we find a particular partition of the vertex set of claw-free strongly chordal graphs in which each element is a clique, and we show that the adjacency graph of these cliques is a tree. In particular, the presented results imply the existence of an ordering of the vertices, and a corresponding edge orientation, such that each directed path is contained in at most two maximal cliques. As shown by the authors in previous works, this allows to give performance guarantee approximation results on a wide class of optimization problems.
2 citations
01 Jan 1999
TL;DR: In this paper, the problem of matching two trees by constructing the association graph using the graph-theoretic concept of connectivity is solved using replicator dynamical systems developed in theoretical biology.
Abstract: It is well-known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a (derived) "association graph." However, it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized, i.e., are trees, since maximal cliques are not constrained to preserve the partial order. Here, we provide a solution to the problem of matching two trees by constructing the association graph using the graph-theoretic concept of connectivity. We prove that, in the new formulation, there is a one-to-one correspondence between maximal cliques and maximal subtree isomorphisms. This allows us to cast the tree matching problem as an indefinite quadratic program using the Motzkin- Straus theorem, and we use "replicator" dynamical systems developed in theoretical biology to solve it. Such continuous solutions to discrete problems are attractive because they can motivate analog and biological implementations. The framework is also extended to the matching of attributed trees by using weighted association graphs. We illustrate the power of the approach by matching articulated and deformed shapes described by shock trees. Index Terms—Maximal subtree isomorphisms, association graphs, maximal cliques, replicator dynamical systems, shock trees, shape recognition.
2 citations
2 citations