α-degree closures for graphs

Bondy and Chvátal (1976) [7] introduced a general and unified approach to a variety of graph-theoretic problems. They defined the k -closure C k ( G ) , where k is a positive integer, of a graph G of order n as the graph obtained from G by recursively joining pairs of nonadjacent vertices a , b satisfying the condition C ( a , b ) : d ( a ) + d ( b ) ≥ k . For many properties P , they found a suitable k (depending on P and n ) such that C k ( G ) has property P if and only if G does. For instance, if P is the Hamiltonian property, then k = n . ouche and Christofides (1987) [5], we proved that C ( a , b ) can be replaced by d ( a ) + d ( b ) + | Q ( G ) | ≥ k , where Q ( G ) is a well-defined subset of vertices nonadjacent to a , b . ouche and Christofides (1981) [4], we proved that, for a ( 2 + k − n ) -connected graph, C ( a , b ) can be replaced by | N ( a ) ∪ N ( b ) | + δ a b + ε a b ≥ k , where ε a b is a well-defined binary variable and δ a b is the minimum degree over all vertices distinct from a , b and nonadjacent to them. The condition on connectivity is a necessary one. s paper we show that C ( a , b ) can be replaced by the condition d ( a ) + d ( b ) + ( α ¯ a b − α a b ) ≥ k , where α ¯ a b and α a b are respectively the order and the independence number of the subgraph G − N ( a ) ∪ N ( b ) . last three conditions are all uncomparable, unique and well-defined. Moreover any Hamiltonian cycle in C n ( G ) can be transformed into a Hamiltonian cycle in the original graph within a polynomial time. However, unlike the conditions given in Ainouche (in preparation) [3] and Ainouche and Christofides (1981) [4], the condition ( α ¯ a b − α a b ) cannot be computed in polynomial time. By giving suitable upper bounds of α a b (or lower bounds of ( α ¯ a b − α a b t ) ) we satisfy this last nice property. In doing so, we surprisingly obtain a result of Broersma and Schiermeyer (1994) [9] as an easy corollary.