Representing Graphs via Pattern Avoiding

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 = {1, . . . , n}, this is equivalent to saying that G is word-representable if for all x, y ∈ {1, . . . , n}, xy ∈ E if and only if the subword w{x,y} of w consisting of all occurrences of x or y in w has no consecutive occurrence of the pattern 11. In this paper, we introduce the study of u-representable graphs for any word u ∈ {1, 2}. A graph G is u-representable if and only if there is a vertex-labeled version of G, G = ({1, . . . , n}, E), and a word w ∈ {1, . . . , n} such that for all x, y ∈ {1, . . . , n}, xy ∈ E if and only if w{x,y} has no consecutive occurrence of the pattern u. Thus, word-representable graphs are just 11-representable graphs. We show that for any k ≥ 3, every finite graph G is 1representable. 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 cointerval graphs and permutation graphs. We also state a number of facts on 12-representation of induced subgraphs of a grid graph.