Bridge and cycle degrees of vertices of graphs. (1984) - Chartrand, Gary; Saba, Farrokh; Wormald, Nicholas C.
Abstract : The bridge degreebdeg   v and cycle degreecdeg   v of a vertexv in a graphG are, respectively, the number of bridges and number of cycle edges incident withv inG . A characterization of finite nonempty setsS of nonnegative integers is given for whichS is the set of bridge degrees (cycle degrees) of the vertices of some graph. The bridge-cycle degree of a vertexv in a graphG is the ordered pair( b, c ), wherebdeg   v = b andcdeg   v = c . Those finite setsS of ordered pairs of nonnegative integers for whichS is the set of bridge-cycle degrees of the vertices of some graph are also characterized.