Representing Graphs via Pattern Avoiding Words

The notion of a word-representable graph has been studied in a series of papers in the literature. A graph G - (V, E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy is an edge in E. If V - {I,...,n}, this is equivalent to saying that C is word-representable if for all x, y is an element of{1,..., n} E if arid only if the subword w({x,y}) consisting of all occurrences of x or y in w has no consecutive occurrence of the pat tern 11. In this paper, we introduce the study of v-representable graphs for any word u is an element of{1,2}*. A graph C is u-representable if and only if there is a vertex-labeled version of G, G ({1,...n}, E), and a word w is an element of{1,...,n}* such that for all x, y E, 01, xy E E if and only if has no consecutive occurrence of the pattern a. Thus, word-representable graphs are just 11-representable graphs. We show that for any k >= 3, every finite graph C is 1(k)-representable. This contrasts with the fact that not all graphs are 11-representable graphs. The main focus of the paper is the study of 12-representable graphs. In particular, we classify the 12-representable trees. We show that any 12-representable graph is a comparability graph and the class of 12-representable graphs include the classes of co-interval graphs and permutation graphs. We also state a number of facts on 12-representation of induced subgraphs of a grid graph.