A note on pseudointersection graphs

A graph C = (V,E ) where V and E are the vertex and edge sets shall be considered to be a simple graph (i.e., finite, undi rected and without loops or multiple edges), and all terms used shall be con· sistent with their definitions in [3J 1. If S is a set and F = {Sl ,S2, . .. , 5 11 } is a family of distinct nonempty subsets of S whose union is 5, then the intersection graph of F, denoted by n (F), is the graph with V (n(F) ) = F such that 5 i and Sj are adjacent if and only if (iff ) i =1= j and 5 i n Sj =1= 0. A graph C is an intersection graph on 5 if there exists such a family F for which C """ n (F). Every graph C is an intersection graph on some finite set [7J , and the intersection number w (C) is the minimum number of elements in a set 5 such that C is an intersection graph on 5. If lSI = n then, as defined by S. Hedetniemi [5J , a representation of C as an intersection graph on S is a one to one function, r :V(C) -'.> {O,l}n, such that for u,v € V(C) one has (u,v) € E(C) iff feU) and r (v) have a 1 in a common coordinate position, and if 1 ::::; i ::::; n then there is some v € V (C) such that r (v) has a 1 in the ith coordinate position. For the complete graph Kg on vertices Vj, V2, and V3 we have W( K 3 ) = 3. If S = {a,b,c} then one can choose, for example, S1 = {a}, S2 = {a,b}, and 53 = {a,c} or 51 = {a,b} , S2 = {b,c} and S3 = {a,c}. In the fo rmer case it is clear that elements band c are needed only to make the S;'s distinct and do nothing to indicate adjacency. Equivalently, for r:V(K 3 ) -'.> {O,lP with r(vd = (1,0,0 ), r (V2) = (1,1,0) and r( vs) = (1,0,1), only the first coordinate has more than one 1 in it. As another example, the graph K4 x is given in fi gure 1 as an intersection graph, and , in thi s case, element c of S is not necessary to indicate the adjacency of any two vertices. The size required for 5 can be reduced by eliminating these" fill ers" used only to obtain distinct representations of each vertex.