Showing papers on "Split graph published in 1976"
TL;DR: The consecutive ones test for the consecutive ones property in matrices and for graph planarity is extended to a test for interval graphs using a recently discovered fast recognition algorithm for chordal graphs.
1,622 citations
TL;DR: An algorithm is presented which finds a maximum independent set in an n-vertex graph in 0($2^{n/3}$) time and can thus handle graphs roughly three times as large as could be analyzed using a naive algorithm.
Abstract: We present an algorithm which finds a maximum independent set in an n-vertex graph in 0($2^{n/3}$) time. The algorithm can thus handle graphs roughly three times as large as could be analyzed using a naive algorithm.
489 citations
01 Nov 1976
TL;DR: In this paper, the maximal size of a clique and the number of Kr's in a complete graph with n points and m edges is investigated. But the maximal clique is not a maximal complete subgraph.
Abstract: Let 0 < p < 1 be fixed and denote by G a random graph with point set , the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables eij, 1 ≤ i < j, defined byare independent r.v.'s with P(eij = 1) = p, P(eij = 0) = 1 − p. Denote by Gn the subgraph of G spanned by the points 1, 2, …, n. These random graphs G, Gn will be investigated throughout the note. As in (1), denote by Kr a complete graph with r points and denote by kr(H) the number of Kr's in a graph H. A maximal complete subgraph is called a clique. In (1) one of us estimated the minimum of kr(H) provided H has n points and m edges. In this note we shall look at the random variablesthe number of Kr's in Gn, andthe maximal size of a clique in Gn.
353 citations
TL;DR: In this paper, it was shown that for any integer r ≥ 1, there is a finite graph H with cl(H ) = m, such that if the edges of H are r -colored in any way, then H contains an induced subgraph G isomorphic to G with all its edges the same color.
153 citations
TL;DR: The resulting system is powerful enough to encompass the algorithms of Tarjan''s type and provide very short proofs on graphs for which the stability number equals the clique-covering number.
Abstract: We formalize certain rules for deriving upper bounds on the stability number of a graph. The resulting system is powerful enough to (i) encompass the algorithms of Tarjan''s type and (ii) provide very short proofs on graphs for which the stability number equals the clique-covering number. However, our main result shows that for almost all graphs with a (sufficiently large) linear number of edges, proofs within our system must have at least exponential length.
51 citations
TL;DR: This paper verifies the truth of the conjecture that all critical graphs have odd order for all graphs of order less than 12 and all graph of order 12 and maximum degree 3 and determines all critical graph through order 7.
44 citations
TL;DR: Using only the set of point-deleted subgraphs, several invariants of a graph are derived and that information is used to show that disconnected graphs and separable graphs without endpoints are reconstructible from their sets of deleted sub graphs.
22 citations
TL;DR: Theorems on directed graphs are used to obtain results on the enumeration of orthogonal space groups in arbitrary dimensions as discussed by the authors, and the graphs themselves provide a convenient notation for the orthorhombic space groups.
Abstract: Theorems on directed graphs are used to obtain results on the enumeration of orthogonal space groups in arbitrary dimensions. In dimension 3 the graphs themselves provide a convenient notation for the orthorhombic space groups.
3 citations