This paper reports work investigating various evolutionary approaches to vertex cover (VC), a well-known NP-Hard optimization problem. Central to each of the algorithms is a novel encoding scheme for VC and related problems that treats each chromosome as a binary decision diagram. As a result, the encoding allows only a (guaranteed optimal) subset of feasible solutions. The encoding also incorp...

A locating-dominating set D of a graph G is dominating where each vertex not in has unique neighborhood D, and the Locating-Dominating Set problem asks if contains such bounded size. This known to be NP-hard even on restricted classes, as interval graphs, split planar bipartite subcubic graphs. On other hand, it solvable polynomial time for some trees and, more generally, graphs cliquewidth. Wh...

In 1994 S. McGuinness showed that any greedy clique decomposition of an n-vertex graph has at most ⌊n2/4⌋ cliques (The greedy clique decomposition of a graph, J. Graph Theory 18 (1994) 427-430), where a clique decomposition means a clique partition of the edge set and a greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empt...

The MaximumWeight Stable Set (MWS) Problem is one of the fundamental problems on graphs. It is well-known to be NP-complete for triangle-free graphs, and Mosca has shown that it is solvable in polynomial time when restricted to P6and triangle-free graphs. We give a complete structure analysis of (nonbipartite) P6and triangle-free graphs which are prime in the sense of modular decomposition. It ...

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...

For an arbitrary graph G = (V,E), let X be a graphical property that can be possessed, or satisfied by the subsets of V . For example, being a clique (maximal complete subgraph), a maximal independent set, an edge, a closed neighborhood, a minimal dominating set, etc. Let CX ={A|A ⊆ V , |A| > 1, and A possesses or satisfies property X}. A set S is an X cover (or X free) if A ∩ S (or A ∩ (V − S)...

A graph G = (V,E) is a unipolar graph if there exits a partition V = V1 ∪ V2 such that, V1 is a clique and V2 induces the disjoint union of cliques. The complement-closed class of generalized split graphs are those graphs G such that either G or the complement of G is unipolar. Generalized split graphs are a large subclass of perfect graphs. In fact, it has been shown that almost all C5-free (a...

This paper extends the recently introduced Phased Local Search (PLS) maximum clique algorithm to unweighted / weighted maximum independent set and minimum vertex cover problems. PLS is a stochastic reactive dynamic local search algorithm that interleaves sub-algorithms which alternate between sequences of iterative improvement, during which suitable vertices are added to the current sub-graph, ...

A graph G is well-covered if every maximal independent set has the same cardinality q. Let ik(G) denote the number of independent sets of cardinality k in G. Brown, Dilcher, and Nowakowski conjectured that the independence sequence (i0(G), i1(G), . . . , iq(G)) was unimodal for any well-covered graph G with independence number q. Michael and Traves disproved this conjecture. Instead they posite...