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- W2913866635 abstract "We introduce a new class of graphs which we call round graphs. These are those graphs whose vertices can be circularly enumerated so that either the neighbourhood or the closed neighbourhood of each vertex forms an interval in the enumeration. We show that round graphs have nice structural properties. We identify two subclasses of round graphs, namely, round graphs of type $I$ and round graphs of type $II$. These two classes can be transformed to each other by taking the complementary graphs. We observe that, by a result of Tucker cite{matrixchar}, round graphs of type $I$ form a class of graphs, which properly contains the class of proper circular arc graphs and is properly contained in the class of general circular arc graphs. We show that round graphs of type $I$ (and hence of type $II$) can be recognized in $O(n+m)$ time (here $n$ denotes the number of vertices and $m$ the number of edges of the graph in question). We show how to reduce the problem of recognizing general round graphs to the problem of determining whether a weakly triangulated graph is round (a problem which is still open). We describe optimal $O(n+m)$ time algorithms for finding a maximum clique, an maximum matching, and a hamiltonian cycle (if one exists), for the class of round graphs of type $II$. Finally, we pose a number of open problems and conjectures concerning the structure and algorithmic properties of round graphs." @default.
- W2913866635 created "2019-02-21" @default.
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- W2913866635 date "1997-05-13" @default.
- W2913866635 modified "2023-10-18" @default.
- W2913866635 title "Round Graphs" @default.
- W2913866635 hasPublicationYear "1997" @default.
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