Discrete Mathematics 115 (1993) 51–55 Graphs with no induced C 4 and 2K 2 Zolt´ anBl´azsik Mih´ aly Hujter Andr´asPluh´ar Zsolt Tuza Abstract. We characterize the structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph. This self-complementary class G of graphs includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set. In the members of G , the number of cliques (as well as the number of maximal independent sets) cannot exceed the number of vertices. Moreover, these graphs are almost extremal to the theorem of Nordhaus and Gaddum (1956). 1. Results. We study undirected graphs without loops and multiple edges. A graph G is called F -free if no induced subgraph of G is isomorphic to F. In this note we give the structural characterization of the self-complementary class G of graphs which are C 4 -free and 2K 2 -free, where C 4 denotes the cycle of four vertices and 2K 2 is the matching of two edges, i.e., the complement of C 4 . Theorem 1.1. A graph G =(V,E) is C 4 -free and 2K 2 -free if and only if there is a partition V 1 ∪ V 2 ∪ V 3 = V with the following properties (i) V 1 is an independent set in G. (ii) V 2 is the vertex set of a complete subgraph in G. (iii) V 3 = ∅ or |V 3 | =5, and in the latter case V 3 induces a 5-cycle in G. (iv) If V 3 = ∅, then for all v i ∈ V i , i =1, 2, 3, v 1 v 3 ∈ E and v 2 v 3 ∈ E hold. A split graph, as introduced in [3, 5], is a graph satisfying properties (i) and (ii) of Theorem 1.1 with V 1 ∪ V 2 = V . Hence, G is a natural extension of the thoroughly investigated class of split graphs. On the other hand, G is a subclass of pseudothreshold graphs characterized by Chv´ atal and Hammer in [1, Theorem 4(iii)] as the graphs G =(V,E) admitting a vertex partition V 1 ∪ V 2 ∪ V 3 = V that satisfies (i), (ii), (iv), and the further property that no three vertices in V 3 are pairwise non-adjacent. Corollary 1.2. If G =(V,E) is a C 4 -free and 2K 2 -free graph, then: (i) either G is a split graph, or there are exactly five distinct vertices v i ∈ V , i =1,..., 5, such that each G − v i is a split graph, (ii) G is a pseudothreshold graph. Recently, Pr¨ omel and Steger [7] proved the following closely related asymp- totic result: the ratio of the numbers of split graphs and C 4 -free graphs on n vertices tends to 1 as n tends to infinity. 1