The square of a block graph

AbstractThe square H2 of a graph H is obtained from H by adding new edges between every two vertices having distance two in H. A block graph is one in which every block is a clique. For the first time, good characterizations and a linear time recognition of squares of block graphs are given in this paper. Our results generalize several previous known results on squares of trees

Algebraic theory for the clique operator

In this text we attempt to unify many results about the K operator based on a new theory involving graphs, families and operators. We are able to build an "operator algebra" that helps to unify and automate arguments. In addition, we relate well-known properties, such as the Helly property, to the families and the operators. As a result, we deduce many classic results in clique graph theory from the basic fact that CS = I for conformal, reduced families. This includes Hamelink's construction, Roberts and Spencer theorem, and Bandelt and Prisner's partial characterization of clique-fixed classes [2]. Furthermore, we show the power of our approach proving general results that lead to polynomial recognition of certain graph classes.Facultad de Ciencias Exacta

A Weakly-Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a {\em weakly robust} polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) that it computes a clique partition, we show that it is guaranteed to be within (1+\eps) ratio of the optimum if the input is UDG; however if the input is not a UDG it either computes a clique partition as in case (i) with no guarantee on the quality of the clique partition or detects that it is not a UDG. Noting that recognition of UDG's is NP-hard even if we are given edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be transformed into an O(\frac{\log^* n}{\eps^{O(1)}}) time distributed PTAS. We consider a weighted version of the clique partition problem on vertex weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas useful in obtaining a PTAS breakdown. Yet, surprisingly, it admits a (2+\eps)-approximation algorithm for the weighted case where the graph is expressed, say, as an adjacency matrix. This improves on the best known 8-approximation for the {\em unweighted} case for UDGs expressed in standard form.Comment: 21 pages, 9 figure

Hierarchical clique analysis in social networks due to common knowledge of proverbs.

24th European Conference on Operational Research (EURO XXIV). Lisboa, 11 a 14 de Julho de 2010 (Comunicação).We present the Hierarchical Clique Analysis, a new algorithm for social networks analysis. The algorithm is exemplified with data about the recognition of proverbs collected in interviews in all Azorean islands and also in three Azorean emigration locations in the USA. Interpreting the set of this data as an incidence matrix of a graph, we obtain 8 oriented and isolated sub-graphs which distinguish the society in a kind of different families of proverbial users. The Hierarchical Clique Analysis finds distinct clusters with a high inner homogeneity

Hierarchical clique analysis in social networks due to common knowledge of proverbs.

24th European Conference on Operational Research (EURO XXIV). Lisboa, 11 a 14 de Julho de 2010 (Comunicação).We present the Hierarchical Clique Analysis, a new algorithm for social networks analysis. The algorithm is exemplified with data about the recognition of proverbs collected in interviews in all Azorean islands and also in three Azorean emigration locations in the USA. Interpreting the set of this data as an incidence matrix of a graph, we obtain 8 oriented and isolated sub-graphs which distinguish the society in a kind of different families of proverbial users. The Hierarchical Clique Analysis finds distinct clusters with a high inner homogeneity

Perfect Graphs

This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement