Absolute retracts of split graphs Original Research Article

It is proved that a split graph is an absolute retract of split graphs if and only if a partition of its vertex set into a stable set and a complete set is unique or it is a complete split graph. Three equivalent conditions for a split graph to be an absolute retract of the class of all graphs are given. It is finally shown that a reflexive split graph G is an absolute retract of reflexive split graphs if and only if G has no retract isomorphic to some Jn, n⩾3. Here Jn is the reflexive graph with vertex set {x1,x2,…, xn, y1,y2,…, yn} in which the vertices x1,x2,…, xn are mutually adjacent and the vertex yi is adjacent to x1,x2,…, xi−1,xi+1,…, xn.