, Then a ?-set of G has size three. Now {v 1 , w 2 , v 3 } and {w 1 , v 2 , w 3 } are two disjoint ?-sets of G

, Hence the Twin Clique Partition of G contains no vertex in core

, A (claw, P 7 )-free graph G. The black vertex is in core(G) ? V 0, Figure, vol.6

, ) be a connected (claw, bull)-free graph with at least two vertices. v ? core(G) if and only if ?(G ? v) > ?(G)

, Hence from Lemma 3.1 there is an induced cycle C k , k ? 6, that contains v. Let the set of vertices of C k be C k = {v 1 , v 2 , . . . , v k }. If V = C k then core(G) = ?. So it exists w ? V ? C k and an edge wu, u ? C k . W.l.o.g let u = v 1 . If wv 2 , wv k are two non-edges then G[{w, v 1 , v 2 , v k }] is a claw. If w has five neighbors in C k the G has a claw. If w has exactly two (successive) neighbors in C k, G (corresponding to the reduced graph of T CP (G)) ? ? a ? v ? b ? ? is an induced path P and it exists ?, vol.4

, corona(G)?core(G), anticore(G) and V 0 , V + , we give connected graphs (without isolated vertices) whose vertex-set correspond to a specific partition. Two of them answer to two open questions by V

E) a connected graph with at least two vertices for which V = V ? is given in, p.139 ,

, Theorem 5.23): these graphs must be such that core(G) = ?, the complete bipartite graph K 3,3 is one of them. Authors showed that graphs with core(G) = ? can exist but no such graph is exhibited. Finding such a graph correspond to the first question in the following article, p.147

On domination and independent domination numbers of a graph, Discrete Mathematics, vol.23, pp.73-76, 1978. ,

Domination alteration sets in graphs, Discrete Mathematics, vol.47, pp.153-161, 1983. ,

Dominating sets for split and bipartite graphs, Information Processing Letters, vol.19, pp.37-40, 1984. ,

, Graphs Theory, 2008.

Dominating sets in chordal graphs, SIAM J. Comput, vol.11, issue.1, pp.191-199, 1982. ,

On the number of vertices belonging to all maximum stable sets of a graph, Discrete Applied Mathematics, vol.124, issue.1, pp.17-25, 2002. ,

, Graph Classes: A survey, 2004.

A linear algorithm for the domination number of a tree, Information Processing Letters, vol.4, issue.2, pp.41-44, 1975. ,

Slater Fundamentals of Domination in Graphs, 1998. ,

Forbidden subgraphs and the Hamiltonian theme, The Theory and Applications of Graphs, pp.297-316, 1981. ,

Vertices Contained in Every Minimum Dominating Set of a Tree, J. Graph Theory, vol.31, pp.163-177, 1999. ,

Changing and unchanging of the domination number of a graph, Discrete Mathematics, vol.308, pp.5015-5025, 2008. ,