Representations of graphs and networks (coding, layouts and embeddings)

Kozyrev, V. P.; Yushmanov, S. V.

doi:10.1007/bf01097528

Cited by 2 publications

(2 citation statements)

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“…, x 2n be consecutive points around a circle, and we draw a chord named a between Circle graphs can also be geometrically represented as overlap graphs of intervals. See the survey by Kozyrev and Yushmanov [28] and the books by Golumbic [24] and Spinrad [36]. The representation of a circle graph as a set of chords is intuitively clear, but the one using a double occurrence word is more convenient for formal proofs.…”

Section: Circle Graphsmentioning

confidence: 99%

“…If a letter represents a chord, a set of chords of a circle is a word corresponding to the sequence of extremities of chords read around the circle, and the chords represented by a and b intersect if and only if the word can be written aubvawbx for some words u, v, w, x. Other characterizations are reviewed in the survey article by Kozyrev and Yushmanov [28] and in the book by Spinrad [36].…”

Section: Introductionmentioning

confidence: 98%

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Circle graphs and monadic second-order logic

Courcelle

2008

Journal of Applied Logic

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This article is part of a project consisting in expressing, whenever possible, graph properties and graph transformations in monadic second-order logic or in its extensions using modulo p cardinality set predicates or auxiliary linear orders. A circle graph is the intersection graph of a set of chords of a circle. Such a set is called a chord diagram. It can also be described by a word with two occurrences of each letter, called a double occurrence word. If a circle graph is prime for the split (or join) decomposition defined by Cunnigham, it has a unique representation by a chord diagram, and this diagram can be defined by monadic secondorder formulas with the even cardinality set predicate. By using the (canonical) split decomposition of a circle graph, we define in monadic second-order logic with auxiliary linear orders all its chord diagrams. This construction uses the fact that the canonical split decomposition of a graph can be constructed in monadic second-order logic with help of an arbitrary linear order. We prove that the order of first occurrences of the letters in a double occurrence word w that represents a connected circle graph determines this word in a unique way. The word w can be defined by a monadic second-order formula from the word of first occurrences of letters. We also prove that a set of circle graphs has bounded clique-width if and only if all the associated chord diagrams have bounded tree-width.

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Section: Circle Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Circle graphs and monadic second-order logic

Courcelle

2008

Journal of Applied Logic

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Convex Polygon Intersection Graphs

Leeuwen

2011

Graph Drawing

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Abstract. Geometric intersection graphs are graphs determined by intersections of geometric objects. We study the complexity of visualizing the arrangements of objects that induce such graphs. We give a general framework for describing geometric intersection graphs, using arbitrary finite base sets of rationally given convex polygons and affine transformations. We prove that for every class of intersection graphs that fits the framework, the graphs in the class have a representation using polynomially many bits. Consequently, the recognition problem of these classes is in NP (and thus NP-complete). We also give an algorithm to find a drawing of the objects in the plane, if a graph class fits the framework.

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Representations of graphs and networks (coding, layouts and embeddings)

Cited by 2 publications

References 219 publications

Circle graphs and monadic second-order logic

Circle graphs and monadic second-order logic

Convex Polygon Intersection Graphs

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