Abstract
A circle graph is the intersection graph of a family of chords on a circle. There is no known characterization of circle graphs by forbidden induced subgraphs that do not involve the notions of local equivalence or pivoting operations. We characterize circle graphs by a list of minimal forbidden induced subgraphs when the graph belongs to one of the following classes: linear domino graphs, P 4 -tidy graphs, and tree-cographs. We also completely characterize by minimal forbidden induced subgraphs the class of unit Helly circle graphs, which are those circle graphs having a model whose chords have all the same length, are pairwise different, and satisfy the Helly property.
Highlights
All graphs in this work are undirected, without multiple edges and without loops
We present some results in this direction, providing forbidden induced subgraph characterizations of circle graphs within different graph classes
Every nontrivial cograph is either disconnected or the join of two smaller cographs. (This fact was discovered independently by several authors since the 1970s; early references include [27].) We are interested in the characterization of circle graphs within two superclasses of cographs: P4-tidy graphs and tree-cographs
Summary
All graphs in this work are undirected, without multiple edges and without loops. Let G be a graph, with vertex set V (G) and edge set E(G). Bouchet gave the following characterization of circle graphs in terms of forbidden induced subgraphs and local equivalence. G is a circle graph if and only if no graph locally equivalent to G contains W5, W7, or BW3 as an induced subgraph (see Fig. 2). The list of 33 minimal forbidden induced subgraphs for this class is obtained using a computer, closing under local complementation the graphs W5, W7 and BW3. In spite of the mentioned works, there are not known characterizations of circle graphs only by forbidden induced subgraphs, i.e., not involving the notions of local equivalence or pivoting operations. In the last section, we completely characterize by minimal forbidden induced subgraphs the class of unit Helly circle graphs, which are those circle graphs having a model whose chords have all the same length, are pairwise different, and satisfy the Helly property. For definitions and notions not introduced and used throughout the paper, the reader is referred to [5]
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