, 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

G. , 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

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