Mutant knots and intersection graphs

We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. The converse statement is easy and well known. We discuss relationship between our results and certain Lie algebra weight systems.Comment: 13 pages, many figure

Enumerating the maximal cliques of a circle graph

We describe the notion of locally transitive orientations of an undirected graph, as a generalization of ordinarytransitive orientations. As an application, we obtain an algorithm for generating all maximal cliquesof a circle graph G in time 0(n(m+α)), where n,m and a are the number of vertices, edges and maximal cliques of G. In addition, we show that the actual number of such cliques can be computedin 0(nm) time.Descrevemos a noção de orientação localmente transitiva de um grafo não direcionado, como uma generalização de orientação transitiva ordinária. Com aplicação obtemos um algoritmo para a geração de todas as cliques maximais de um grafo circular G, cuja complexidade é 0(n(m+α)), onde n, m e a são o número de vértices, arestas e cliques maximais de G. Em adição, mostraremos que o número exato de tais cliques pode ser computado em tempo 9(nm)

Automorphism Groups of Geometrically Represented Graphs

We describe a technique to determine the automorphism group of a geometrically represented graph, by understanding the structure of the induced action on all geometric representations. Using this, we characterize automorphism groups of interval, permutation and circle graphs. We combine techniques from group theory (products, homomorphisms, actions) with data structures from computer science (PQ-trees, split trees, modular trees) that encode all geometric representations. We prove that interval graphs have the same automorphism groups as trees, and for a given interval graph, we construct a tree with the same automorphism group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982]. For permutation and circle graphs, we give an inductive characterization by semidirect and wreath products. We also prove that every abstract group can be realized by the automorphism group of a comparability graph/poset of the dimension at most four