Finding clusters of well-connected nodes in a graph is a problem common to many domains, including social networks, the Internet and bioinformatics. From a computational viewpoint, finding these clusters or graph communities is a difficult problem. We use a clique matrix decomposition based on a statistical description that encourages clusters to be well connected and few in number. The formal ...

We answer a question of Brandstädt et al. by showing that deciding whether a line graph with maximum degree 5 has a stable cutset is NP -complete. Conversely, the existence of a stable cutset in a line graph with maximum degree at most 4 can be decided efficiently. The proof of our NP -completeness result is based on a refinement on a result due to Chvátal that recognizing decomposable graphs w...

This survey emphasises results in graph theory which were fairly difficult to prove. Because it is a personal view of the subject, this section will describe three of my own results which I personally found fairly difficult to prove (whatever their degree of difficulty in an absolute sense may be). A decomposition of a graph G is a set of subgraphs of G such that each edge of G belongs to exact...

Given a graph G $G$ , decomposition of is partition its edges. A ( d h ) $(d,h)$ -decomposable if edge set can be partitioned into $d$ -degenerate and with maximum degree at most $h$ . For ≤ 4 $d\le 4$ we are interested in the minimum integer ${h}_{d}$ such that every planar $(d,{h}_{d})$ -decomposable. It was known 3 ${h}_{3}\le 2 8 ${h}_{2}\le 8$ 1 = ∞ ${h}_{1}=\infty $ This paper proves ${h}...

This paper makes two contributions to the computational geometry of decomposable graphs, aimed primarily at facilitating statistical inference about such graphs where they arise as assumed conditional independence structures in stochastic models. The first of these provides sufficient conditions under which it is possible to completely connect two disconnected cliques of vertices, or perform th...

This thesis consists of four papers studying structure learning and Bayesian inference in probabilistic graphical models for both undirected and directed acyclic graphs (DAGs). Paper A presents a novel algorithm, called the Christmas tree algorithm (CTA), that incrementally construct junction trees for decomposable graphs by adding one node at a time to the underlying graph. We prove that CTA w...

A graph G is locally irregular if every two adjacent vertices of G have different degrees. A locally irregular decomposition of G is a partition E1, ..., Ek of E(G) such that each G[Ei] is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, in that sense, it was conjectured by Baudon, Bensmail, Przyby lo and Woźniak that they decompose i...

A walk on an edge-colored graph G is a path in G which contains each edge at least once, and the trace of a walk w is the sequence of edge-colors seen in w. We investigate the walk realizability problem, which is to decide, given an edge-colored graph G and a string x of colors, if there is a walk on G with trace x. We de ne a method of decomposing graphs into linear chains, and show that the n...

He, Hou, Lih, Shao, Wang, and Zhu showed that a planar graph of girth 11 can be decomposed into a forest and a matching. Borodin, Kostochka, Sheikh, and Yu improved the bound on girth to 9. We give sufficient conditions for a planar graph with 3-cycles to be decomposable into a forest and a matching. c © 2008 Elsevier B.V. All rights reserved.