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In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by Földes and Hammer ( 1977a , 1977b ), and independently introduced by Tyshkevich and Chernyak ( 1979 ), where they called these graphs "polar graphs" ( Russian ...
In graph theory, a split of an undirected graph is a cut whose cut-set forms a complete bipartite graph. A graph is prime if it has no splits. The splits of a graph can be collected into a tree-like structure called the split decomposition or join decomposition , which can be constructed in linear time.
2. A split of an arbitrary graph is a partition of its vertices into two nonempty subsets, such that the edges spanning this cut form a complete bipartite subgraph. The splits of a graph can be represented by a tree structure called its split decomposition. A split is called a strong split when it is not crossed by any other split.
Word-representation of split graphs is studied in. [22] [13] In particular, [22] offers a characterisation in terms of forbidden induced subgraphs of word-representable split graphs in which vertices in the independent set are of degree at most 2, or the size of the clique is 4, while a computational characterisation of word-representable split ...
In graph theory, a branch of mathematics, the splittance of an undirected graph measures its distance from a split graph. A split graph is a graph whose vertices can be partitioned into an independent set (with no edges within this subset) and a clique (having all possible edges within this subset). The splittance is the smallest number of edge ...
Optimal coloring of a split graph, obtained by giving each vertex of a maximal clique (heavy vertices and edges) a separate color, and then giving each remaining vertex the same color as a clique vertex to which it is not adjacent. A split graph is a graph that can be partitioned into a clique and an independent set. It can be colored by ...
When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.
In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the ...