A connectivity game for graphs

The introduction of a {0,1}-valued game associated to a connected graph allows us to assign to each vertex a value of weighted connectivity according to the different solutions that for cooperative games are obtained by means of semivalues. The marginal contributions of each vertex to the coalitions differentiate an active connectivity from another reactive connectivity, according to whether the vertex is essential to obtain the connection or is the obstacle for the connection between the vertices in the coalition. We offer general properties of the connectivity, as well as the behaviour of different families of graphs with regard to this concept. We also analyse the effect on different vertices due to the addition of an edge to the initial graph. Copyright Springer-Verlag 2004