Not every edge-clique graph is a clique graph.

An i-triangulated graph is a graph in which every odd cycle has two non-crossing chords; i-triangulated graphs form a subfamily of perfect graphs. A slightly more general family of perfect graphs are clique-separable graphs. A graph is clique-separable precisely if every induced subgraph either has a clique cutset, or is a complete multipartite graph or a clique joined to an arbitrary bipartite...

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. We have recently shown that any greedy clique decomposition of a graph of order n has at most-~ cliques. A greedy max-clique decomposition is a particular kind of greedy clique decomposition where maximum cliques are removed, instead of just maximal ones. In this pa...

The clique graph K(G) of a graph G is the intersection graph of all its (maximal) cliques. We explore the effect of operations like edge contraction, edge removal and others on the dynamical behaviour of a graph under the iteration of the clique operator K. As a consequence of this study, we can now prove the clique divergence of graphs for which no previously known technique would yield the re...

Peer cliques form an important context for the social development of adolescents. Although clique members are often similar in social status, also within cliques, status differences exist. How differences in social status between clique members are related to behaviors of its individual members is rather unknown. This study examined to what extent the relationship of individual social status (i...

Bayesian networks (BNs) are used to represent and efficiently compute with multi-variate probability distributions in a wide range of disciplines. One of the main approaches to perform computation in BNs is clique tree clustering and propagation. In this approach, BN computation consists of propagation in a clique tree compiled from a Bayesian network. There is a lack of understanding of how cl...

A clique-colouring of a graph G is a colouring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, H(G), of a graph G has V (G) as its set of vertices and the maximal cliques of G as its hyperedges. A vertex-colouring of H(G) is a clique-colouring of G. Determining the clique-chromatic number, the least number of colours for which a graph...

The clique partitioning problem (CPP) can be formulated as follows: Given is a complete graph G = (V, E), with edge weights wij ∈ R for all {i, j} ∈ E. A subset A ⊆ E is called a clique partition if there is a partition of V into nonempty, disjoint sets V1, . . . , Vk , such that each Vp (p = 1, . . . ,k) induces a clique (i.e., a complete subgraph), and A = ∪ p=1 {{i, j}|i, j ∈ Vp , i ≠ j}. Th...

We offer an algorithm that finds a clique tree such that the size of the largest clique is at most (2α + 1)k where k is the size of the largest clique in a clique tree in which this size is minimized and α is the approximation ratio of an α-approximation algorithm for the 3-way vertex cut problem. When α = 4/3, our algorithm’s complexity is O(24.67kn · poly(n)) and it errs by a factor of 3.67 w...

Daligault, Rao and Thomassé proposed in 2010 a fascinating conjecture connecting two seem-ingly unrelated notions: clique-width and well-quasi-ordering. They asked if the clique-width ofgraphs in a hereditary class which is well-quasi-ordered under labelled induced subgraphs is boundedby a constant. This is equivalent to asking whether every hereditary class of unbounded clique-...