We introduce a special decomposition, the so-called split-minors, of the reduced clique graphs of chordal graphs. Using this notion, we characterize asteroidal sets in chordal graphs and clique trees with minimum number of leaves.

‎The Narumi-Katayama index was the first topological index defined‎ ‎by the product of some graph theoretical quantities‎. ‎Let $G$ be a ‎simple graph with vertex set $V = {v_1,ldots‎, ‎v_n }$ and $d(v)$ be‎ ‎the degree of vertex $v$ in the graph $G$‎. ‎The Narumi-Katayama ‎index is defined as $NK(G) = prod_{vin V}d(v)$‎. ‎In this paper,‎ ‎the Narumi-Katayama index is generalized using a $n$-ve...

The clique number of an undirected graphG is the maximum order of a complete subgraph of G and is a well-known lower bound for the chromatic number ofG. Every proper k-coloring of G may be viewed as a homomorphism (an edge-preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natu...

Many important problems arising in computational biochemistry and genomics have been formulated in terms of underlying combinatorial optimization models. In particular, a number have been formulated as clique-detection models. The proposed article includes an introduction to the underlying biochemistry and genomic aspects of the problems as well as to the graph-theoretic aspects of the solution...

This is a survey of the results in [1, 2, 3, 4, 5, 6] on convexity numbers of closed sets in Rn, homogeneity numbers of continuous colorings on Polish spaces and families of functions covering Rn. 1. Convexity numbers of closed sets in R A natural way to measure the degree of non-convexity of a subset S of a real vector space is the convexity number γ(S), the least size of a family C of convex ...

The Lescure–Meyniel conjecture is the analogue of Hadwiger’s for immersion order. It states that every graph G contains complete Kχ(G) as an immersion, and like its minor-order counterpart it open even graphs with independence number 2. We show α(G)≥2 no hole length between 4 2α(G) satisfies this conjecture. In particular, C4-free α(G)=2 give another generalisation corollary, follows. Let H be ...

Let ! be the induced-minor relation. It is shown that, for every t, all chordal graphs of clique number at most t are well-quasi-ordered by !. On the other hand, if the bound on clique number is dropped, even the class of interval graphs is not well-quasi-ordered by !. c © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 105–114, 1998

If a graph has no induced subgraph isomorphic to any graph in a finite family {H1, . . . ,Hp}, it is said to be (H1, . . . ,Hp)-free. The class of H-free graphs has bounded clique-width if and only if H is an induced subgraph of the 4-vertex path P4. We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs H1 and H2. Prior to our study it was ...

Numerous problems can be modeled as clique partitioning problems in digital design synthesis. In this paper, we present two new polynomial time heuristic algorithms for efficient clique partitioning with or without limiting the maximum clique size. The goal of clique partitioning is to partition a graph into a minimum number of cliques. The basic approach of the new algorithm is to find small c...