Abstract
In this paper we present a new method for studying graphs. Generally speaking this involves decomposing a graph into two disjoint subgraphs which are connected by special sets of lines. We consider four types of connections between these subgraphs, i.e., those for which the set of connecting lines describes a function, a homomorphism, a permutation, or an automorphism.
We consider this manner of decomposing a graph to be useful for studying a wide variety of parameters and properties of graphs. To illustrate this we obtain results relating to such concepts as arboricity, thickness, biparticity, and chromatic number. We derive a method for constructing new classes of critical graphs and obtain several isomorphism theorems for classes of permutation graphs, one of which involves the group theoretic concept of a double coset.
Research supported in part by the Office of Naval Research, Contract Number NR 043-367.
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References
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Hedetniemi, S. (1969). On classes of graphs defined by special cutsets of lines. In: Chartrand, G., Kapoor, S.F. (eds) The Many Facets of Graph Theory. Lecture Notes in Mathematics, vol 110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0060115
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DOI: https://doi.org/10.1007/BFb0060115
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