Showing papers on "Split graph published in 1988"
Book•
01 Oct 1988
TL;DR: In this paper, the Spectra of Graphs with Seven Vertices (S7V) with seven vertices is defined. And the Matching Polynomial and Other Graph Polynomials (MOPs) are defined.
Abstract: 1. Characterizations of Graphs by their Spectra. 2. Distance-Regular and Similar Graphs. 3. Miscellaneous Results from the Theory of Graph Spectra. 4. The Matching Polynomial and Other Graph Polynomials. 5. Applications to Chemistry and Other Branches of Science. 6. Spectra of Infinite Graphs. Appendix: Spectra of Graphs with Seven Vertices. Bibliography. Bibliographic Index. Index.
338 citations
TL;DR: The maximum number of maximal independent sets which a connected graph on n vertices can have is determined, and the extremal graphs are completely characterize, thereby answering a question of Wilf.
92 citations
TL;DR: In this paper, the authors provide criteria of isometric embeddability of graphs in Hamming graphs, where the natural metric d(x, y) is defined as the number of edges in the shortest chain between the vertices x and y.
Abstract: In this study, we provide criteria of isometric embeddability of graphs in Hamming graphs We consider ordinary connected graphs with a finite vertex set endowed with the natural metric d(x, y), equal to the number of edges in the shortest chain between the vertices x and y
81 citations
24 Oct 1988
TL;DR: The author gives efficient parallel algorithms for recognizing chordal graphs, finding a maximum clique and a maximum independent set in a chordal graph, and an efficient parallel algorithm for finding a perfect elimination ordering.
Abstract: The author gives efficient parallel algorithms for recognizing chordal graphs, finding a maximum clique and a maximum independent set in a chordal graph, finding an optimal coloring of a chordal graph, finding a breadth-first search tree and a depth-first search tree of a chordal graph, recognizing interval graphs, and testing interval graphs for isomorphism. The key to the results is an efficient parallel algorithm for finding a perfect elimination ordering. >
74 citations
TL;DR: The notion of the boundary clique and the k-overlap clique graph are introduced and the following are proved: Every incomplete chordal graph has two nonadjacent simplicial vertices lying in boundary cliques.
Abstract: We introduce the notion of the boundary clique and the k-overlap clique graph and prove the following: Every incomplete chordal graph has two nonadjacent simplicial vertices lying in boundary cliques. An incomplete chordal graph G is k-connected if and only if the k-overlap clique graph gk(G) is connected. We give an algorithm to construct a clique tree of a connected chordal graph and characterize clique trees of connected chordal graphs using the algorithm.
64 citations
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TL;DR: It is proved that a graph is brittle whenever it is HHD-free, and it follows that an optimal coloring and a largest clique of an H HD-free graph can be found in O(n4) time.
Abstract: Chvatal defined a graph G to be brittle if each induced subgraph F of G contains a vertex that is not a midpoint of any P4 or not an endpoint of any P4. Every brittle graph is perfectly orderable. In this paper, we prove that a graph is brittle whenever it is HHD-free (containing no chordless cycle with at least five vertices, no cycle on six vertices with a long chord, and no complement of the chordless path on five vertices). We also design an O(n4) algorithm to recognize HHD-free graphs, and also an O(n4) algorithm to construct a perfect order of an HHD-free graph. It follows from this result that an optimal coloring and a largest clique of an HHD-free graph can be found in O(n4) time.
55 citations
TL;DR: A complete enumeration is carried out of self-complementary strongly regular graphs with fewer than 53 vertices for new necessary conditions for the cycles of complementing permutations and for the block valencies of the corresponding induced adjacency matrix partitions.
34 citations
TL;DR: An algorithm for finding maximal chordal subgraphs is developed that has worst-case time complexity of O(|E|Δ), where |E| is the number of edges in G and Δ is the maximum vertex degree in G.
33 citations
Dissertation•
01 Jun 1988
TL;DR: The notion of local structure was introduced in this paper, where it was shown that the bipartite, comparability, chordal and chordal graphs do not have local structure.
Abstract: The thesis introduces a property of graph classes called local structure. Intuitively, a graph class has local structure if there exists a constant $c$ such that each graph on $n$ vertices in the class can be encoded in such a way that (1) each vertex is assigned an integer in the range 1,$\dots ,n\sp{c}$. (2) one can decide if vertices $u$ and $v$ are adjacent using only the integers assigned to those two vertices.
One property of local structure is that if a class $C$ has local structure then any class $C\sp\prime$ $\subseteq$ $C$ also has local structure.
Many well known classes of graphs have the local structure property. The bounded degree or degree-k graphs have local structure. The standard adjacency list representation satisfies the first condition above since each vertex has at most k others on its list. Note that this may not be true for an arbitrary graph. In this case adjacency is decided in the obvious way. A related class that also has local structure is the class in which each induced subgraph has at least one vertex of bounded degree, the hereditary degree-k graphs. Planar graphs are all hereditary degree-5 graphs, and trees are all hereditary degree-1 graphs. Thus, both of these classes have local structure. Many geometrically defined classes such as the interval, circle, circular arc and permutation graphs also have local structure.
A counting technique is used to show that classes do not have local structure. A simple version can be used to show that the bipartite, comparability and chordal graphs do not have local structure. A stronger version is required to show that the class in which no graph has more than $\vert V \vert$ edges does not have local structure.
The notion of local structure is related to some questions concerning universal graphs. $G$ is induced-universal for a set $S$ if each graph in $S$ is an induced subgraph of $G$. If a graph class has local structure then for each $n$ there is an induced-universal graph of polynomial size for the graphs in the class on $n$ vertices. If a graph class satisfies a stronger version of local structure then it can be shown that there are induced-universal graphs as above which are also in the class. For example, there is an induced-universal Interval graph of size $O(n\sp3)$ for the interval graphs on $n$ vertices.
30 citations
TL;DR: A class of graphs is introduced that generalize threshold graphs by introducing threshold tolerances, and the complements of these graphs contain interval graphs and threshold graphs, and are contained in the subclass of chordal graphs called strongly chordal graph, and in the class of interval tolerance graphs.
Abstract: In this paper, we introduce a class of graphs that generalize threshold graphs by introducing threshold tolerances. Several characterizations of these graphs are presented, one of which leads to a polynomial-time recognition algorithm. It is also shown that the complements of these graphs contain interval graphs and threshold graphs, and are contained in the subclass of chordal graphs called strongly chordal graphs, and in the class of interval tolerance graphs.
28 citations
TL;DR: The chapter studies the relationships between toughness of a graph and the n-extendability of the graph and proves two results, essentially that if a graph has sufficiently high toughness (and has an even number of points) then it must be n-Extendable.
TL;DR: It is shown that the existence of a Hamilton cycle in the line graph of a graph G can be ensured by imposing certain restrictions on certain induced subgraphs of G, which results in a number of known results on hamiltonian line graphs being improved.
Abstract: It is shown that the existence of a Hamilton cycle in the line graph of a graph G can be ensured by imposing certain restrictions on certain induced subgraphs of G. Thereby a number of known results on hamiltonian line graphs are improved, including the earliest results in terms of vertex degrees. One particular consequence is that every graph of diameter 2 and order at least 4 has a hamiltonian line graph.
TL;DR: If in a clique on n vertices, the edges between cn a vertices are deleted,1/2⩽a≤1, then the number of cliques needed to partition what is left is asymptotic to c 2 n 2a ; this fills in a gap between results of Wallis and Pullman and Donald for a=1, c=1.
TL;DR: The connectedness and clique numbers of the graphs U n are discussed, the infinite graph with n -dimensional rational space Q n as vertex set and two vertices joined by an edge if and only if the distance between them is exactly 1.
TL;DR: Chepoi et al. as discussed by the authors characterized the centers of triangulated graphs with a standard metric d(x, y) equal to the number of edges in the chain of shortest length joining vertexes x, y.
Abstract: CENTERS OF TRIANGULATED GRAPHS V. D. Chepoi Let G = (X, U) be an ordinary graph with arbitrarily (not necessarily finitely) many vertexes, any two of which are joined by some finite chain. We endow G with a standard metric d(x, y) equal to the number of edges in the chain of shortest length joining vertexes x, y. The eccentricity e(z) of a vertex z is defined as {d(z, v): v e X}. The radius r(G) is the least eccentricity of the vertexes, and the diameter d(G) is the largest eccentricity. The center C(G) of G is the subgraph generated by the set of vertexes with minimal eccentricity. It is well known [I, 2] that any graph G, even if it is not connected, is the center of some graph G', i.e., G = C(G'). At the same time, if one confines attention to special classes of graphs, their centers may have rather specific features. Thus, a well-known re- sult of Jordan [3] states that the center of any finite tree consists of one vertex or two adjacent vertexes; that is to say, the only two possibilities are the graphs K l and K 2 (where K n denotes the complete subgraph on n vertexes). The centers of maximal outerplanar graphs and 2-trees were described in [4, 5]. In this paper we characterize the centers of triangu- lated graphs. In this connection, we note that metric properties, including in particular properties of the centers, of triangulated graphs have been studied by various authors [6-9]. Our Theorems 1 and 2 were proved in [8], but they are established here in a more general form and the proofs are simpler. Recall [i0] that a graph G is said to be triangulated if, in any simple cycle F of length more than 3, there are two vertexes not adjacent in r but joined by an edge in G. Recall moreover that a clique of a graph G is any maximal complete subgraph (with re- spect to inclusion). The density of a graph G is the cardinality of the largest clique in G (if such a clique exists). Throughout this paper, the term triangulated graph will mean a triangulated graph of finite diameter without infinite complete subgraphs. Let CT denote the family of graphs which are centers of triangulated graphs. We shall need a number of additional concepts and definitions. Given sets M c X in a graph G and any number r ~ 0, we put where
IBM1
TL;DR: A number of topics in algorithmic combinatorics which involve intersection graphs and their representative families of sets are presented and possible new directions of research are suggested.
Abstract: Let ℛ be a family of sets. The intersection graph of ℛ is obtained by representing each set in ℛ by a vertex and connecting two vertices by an edge if and only if their corresponding sets intersect. Of primary interest are those classes of intersection graphs of families of sets having some specific topological or other structure. The “grandfather” of all intersection graphs is the class of interval graphs, that is, the intersection graphs of intervals on a line.
TL;DR: The algorithm in this paper is based on decomposing these graphs into essentially two special classes of inseparable component graphs whose optimization problems are easy to solve, finding the solutions for these components and combining them to form a solution for the original graph.
Abstract: Efficient decomposition algorithms for the weighted maximum independent set, minimum coloring, and minimum clique cover problems on planar perfect graphs are presented. These planar graphs can also be characterized by the absence of induced odd cycles of length greater than 3 (odd holes). The algorithm in this paper is based on decomposing these graphs into essentially two special classes of inseparable component graphs whose optimization problems are easy to solve, finding the solutions for these components and combining them to form a solution for the original graph. These two classes are (i) planar comparability graphs and (ii) planar line graphs of those planar bipartite graphs whose maximum degrees are no greater than three. The same techniques can be applied to other classes of perfect graphs, provided that efficient algorithms are available for their inseparable component graphs.
Posted Content•
TL;DR: This paper shows that a strongly regular graph with 49 vertices and degree 16, which has parameters (v, k, ?, µ) = (49, 16, 3, 6), cannot exist.
Abstract: Introduction. A strongly regular graph with 49 vertices and degree 16 has parameters (v, k, ?, µ) = (49, 16, 3, 6). In this paper we show that such a graph cannot exist. Until now it was the smallest (with respect to the number of vertices) feasible strongly regular graph for which existence was not settled. Our result is the second 'ad hoc' non-existence result for strongly regular graphs. Earlier , Wilbrink and Brouwer [2) proved that (57, 14, 1, 4) cannot be the parameter set of a strongiy regular graph. At the moment the smallest unsettled case is (65, 32, 15, 16). See Brouwer and Van Lint [1) for a survey of recent results on strongly regular graphs. The present proof involves counting techniques, enumeration, Iinear algebra and the use of a computer. Although only a Iittle computing time was needed, we could not manage without a computer.
01 Jul 1988
TL;DR: The main result is an $O(\log^{2} n) time, $O(n^{6}/\log n)$ processor algorithm for maximum matching on interval graphs, and an improved algorithm for minimum bandwidth on representations of proper interval graphs.
Abstract: In this paper, we consider parallel algorithms on interval graphs. An interval graph is a graph having a one-to-one correspondence with a sequence of intervals on the real line, such that each vertex maps to an interval in the sequence and an edge exists between two vertices if and only if the corresponding intervals overlap. Throughout the paper we use the CREW PRAM model. Our main result is an $O(\log^{2} n)$ time, $O(n^{6}/\log n)$ processor algorithm for maximum matching on interval graphs. We give PT-optimal algorithms for maximum weighted clique, maximum independent set, minimum clique cover, and minimum dominating set for representations of interval graphs; and Hamiltonian circuit for representations of proper interval graphs. We also give an improved algorithm for minimum bandwidth on representations of proper interval graphs. In addition, we present $O (\log n)$ time, $O (n^{2}/\log n)$ processor algorithms for depth-first search on representations of interval graphs and maximum matching on representations of proper interval graphs.
TL;DR: It is proved that this is true for claw-free graphs, those graphs with no induced subgraph isomorphic to K1,3, and this includes line graphs as a special case.
Abstract: The Edge Reconstruction Conjecture states that all graphs with at least four edges are determined by their edge-deleted subgraphs. We prove that this is true for claw-free graphs, those graphs with no induced subgraph isomorphic to K1,3. This includes line graphs as a special case.
TL;DR: It is shown that perfectly orderable graphs are quasi-parity graphs by exhibiting two nodes which are not linked by a chordless odd chain by way of a short and simpler proof.
01 May 1988
TL;DR: It is shown that the RNC-problems UNARY NETWORK FLOW and PERFECT MATCHING becomeP-complete, that the NP-complete problems CHLIQUE, MAXIMUM INDEPENDENT SET and CHROMATIC NUMBER remain NP- complete and that aP- complete version of the CIRCUIT VALUE problem becomes PSPACE-complete.
Abstract: The complexity of graph problems is investigated when the graphs are presented asvertex multiplicity graphs. In this presentation an independent set of vertices which are connected in the same way with the remaining vertices of the graph can be described by giving only one vertex and the size of the independent set in binary. Using this succinct graph presentation one can expect that the complexity of graph problems can blow up exponentially. In fact, it is shown that the RNC-problems UNARY NETWORK FLOW and PERFECT MATCHING becomeP-complete, that the NP-complete problems CHLIQUE, MAXIMUM INDEPENDENT SET and CHROMATIC NUMBER remain NP-complete and that aP-complete version of the CIRCUIT VALUE problem becomes PSPACE-complete.
TL;DR: A polynomial-time algorithm is indicated to recognize a strict 2- threshold graph if its edge-set can be partitioned into two threshold graphs such that every triangle of G is also a triangle of T 1 or of T 2.
TL;DR: In this paper it was shown that if an interval graph possesses a maximal-clique partition then its clique covering and clique partition numbers are equal, and equal to the maximal clique partitions number.
Abstract: It is shown that if an interval graph possesses a maximal-clique partition then its clique coveringand clique partition numbers are equal, and equal to the maximal-clique partition numberMoreover an interval graph has such a partition if and only if all its maximal cliques areedge-disjoint 1980 Mathematics subject classification (Amer Math Soc): 05 C 35 1 Interval graphs and clique-matricesThroughout this paper graphs are finite, undirected, loopless and without mul-tiple edges A clique is a complete subgraph, and a maximal clique is a cliquewhich is not a proper subgraph of any other cliqueA graph G is called an interval graph if its vertices can be put into one-to-onecorrespondence with a set of intervals 7 of the real line, such that two verticesare connected by an edge of G if and only if the corresponding intervals havenonempty intersection Clearly any induced subgraph of an interval graph is aninterval graphThe earliest characterization of interval graphs was obtained by Lekkerkerkerand Boland [3], as follows
TL;DR: If G is chordal and has maximum clique size ω(G) = m, then i (G) ⩽ [1 + o(1)]m/log2m and this result is best possible, even for split graphs (chordal graphs whose complement is also chordal).
Abstract: The interval number of a (simple, undirected) graph G is the least positive integer t such that G is the intersection graph of sets, each of which is the union of t real intervals. A chordal (or triangulated) graph is one with no induced cycles on 4 or more vertices. If G is chordal and has maximum clique size ω(G) = m, then i(G) ⩽ [1 + o(1)]m/log2m and this result is best possible, even for split graphs (chordal graphs whose complement is also chordal).
TL;DR: The circumstances under which a set of 11 vertices in a 3-connected cubic graph lies on a cycle are determined and the number of such cycles that exist is considered.
Abstract: In this paper we determine the circumstances under which a set of 11 vertices in a 3-connected cubic graph lies on a cycle. In addition, we consider the number of such cycles that exist and characterize those graphs in which a set of 9 vertices lies in exactly two cycles.
21 Dec 1988
TL;DR: This paper extends graph coloring, depth first search and maximal independent sets algorithms to K3,3-free graphs, showing that the restriction of planarity is not important and it is shown that K2,2- free graphs are five colorable.
Abstract: For several problems, restricting attention to special classes of graphs has yielded better algorithms. In particular, restricting to planar graphs yields efficient parallel algorithms for several graph problems. In this paper we extend these algorithms to K3,3-free graphs, showing that the restriction of planarity is not important. The three problems dealt with are : graph coloring, depth first search and maximal independent sets. As a corollary we show that K3,3-free graphs are five colorable (this bound is tight).
01 Jan 1988
TL;DR: In this paper, the existence of a Hamilton cycle in the line graph of a graph G can be ensured by imposing certain restrictions on certain induced subgraphs of G. This was shown to be the case for all graphs of diameter 2 and order at least 4.
Abstract: It is shown that the existence of a Hamilton cycle in the line graph of a graph G can be ensured by imposing certain restrictions on certain induced subgraphs of G. Thereby a number of known results on hamiltonian line graphs are improved, including the earliest results in terms of vertex degrees. One particular consequence is that every graph of diameter 2 and order at least 4 has a hamiltonian line graph.