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A number of generalisations [25] [26] [27] of the notion of a word-representable graph are based on the observation by Jeff Remmel that non-edges are defined by occurrences of the pattern 11 (two consecutive equal letters) in a word representing a graph, while edges are defined by avoidance of this pattern. For example, instead of the pattern ...
If a graph is both a split graph and an interval graph, then its complement is both a split graph and a comparability graph, and vice versa. The split comparability graphs, and therefore also the split interval graphs, can be characterized in terms of a set of three forbidden induced subgraphs. [7] The split cographs are exactly the threshold ...
A circle graph is the intersection graph of a family of chords of a circle. A given graph is a circle graph if and only if each of the quotients of its split decomposition is a circle graph, so testing whether a graph is a circle graph can be reduced to the same problem on the prime quotient graphs of the graph.
A complete bipartite graph with m = 5 and n = 3 The Heawood graph is bipartite.. In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in .
A cut C = (S, T) is a partition of V of a graph G = (V, E) into two subsets S and T. The cut-set of a cut C = (S, T) is the set {(u, v) ∈ E | u ∈ S, v ∈ T} of edges that have one endpoint in S and the other endpoint in T. If s and t are specified vertices of the graph G, then an s – t cut is a cut in which s belongs to the set S and t ...
The monoid of words over an alphabet A is the free monoid over A. That is, the elements of the monoid are the finite sequences (words) of elements of A (including the empty sequence, of length 0), and the operation (multiplication) is the concatenation of words. A word u is a prefix (or 'truncation') of another word v if there exists a word w ...
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A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition. It is a paradigmatic example of a Sturmian word and specifically, a morphic word.