Showing papers on "Split graph published in 1997"
Book•
01 Jan 1997TL;DR: In this paper, the background in graph spectra is described as a background of a graph, and the graph angles and angles of graphs are modeled as Eigenvectors of graphs.
Abstract: 1. A background in graph spectra 2. Eigenvectors of graphs 3. Eigenvectors of techniques 4. Graph angles 5. Angle techniques 6. Graph perturbations 7. Star partitions 8. Canonical star bases 9. Miscellaneous results.
508 citations
TL;DR: A sufficient condition for claw-free graphs, the equivalence of some conjectures on hamiltonicity in 2-tough graphs, and the corresponding conjectures for 7-connected claw free graphs are obtained as corollaries in this paper.
273 citations
TL;DR: It is shown that every connected AT-free graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph.
Abstract: An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triple-free (AT-free) if it contains no asteroidal triples. The motivation for this investigation was provided, in part, by the fact that the AT-free graphs provide a common generalization of interval, permutation, trapezoid, and cocomparability graphs. The main contribution of this work is to investigate and reveal fundamental structural properties of AT-free graphs. Specifically, we show that every connected AT-free graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. We then provide characterizations of AT-free graphs in terms of dominating pairs and minimal triangulations. Subsequently, we state and prove a decomposition theorem for AT-free graphs. An assortment of other properties of AT-free graphs is also provided. These properties generalize known structural properties of interval, permutation, trapezoid, and cocomparability graphs.
156 citations
TL;DR: There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms, and some of the results are surveyed.
Abstract: The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants – see Bollobas [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this paper we survey some of the results in this area.
149 citations
TL;DR: It is shown that computing the bandwidth or interval completion number is NP-hard even for co-bipartite graphs and, on the other side, that there are efficient algorithms for these problems on many other claw-free subclasses of co-comparability graphs.
141 citations
TL;DR: A branch-and-bound algorithm for the maximum clique problem which applies existing clique finding and vertex coloring heuristics to determine lower and upper bounds for the size of amaximum clique is introduced.
136 citations
TL;DR: It is shown that for every fixed real η>0 there exists a constantC=C(η) such that almost every random graphGn, p withp withp=p(n)≥Cn−2/5 satisfies Gn,p→2/3+ηK4.
Abstract: For 0 0 there exists a constantC=C(η) such that almost every random graphGn,p withp=p(n)≥Cn−2/5 satisfiesGn,p→2/3+ηK4 The proof makes use of a variant of Szemeredi's regularity lemma for sparse graphs and is based on a certain superexponential estimate for the number of pseudo-random tripartite graphs whose triangles are not too well distributed Related results and a general conjecture concerningH-free subgraphs of random graphs in the spirit of the Erdős-Stone theorem are discussed
130 citations
Book•
01 Jan 1997
TL;DR: In this paper, Cayley Graphs and Interconnection networks have been studied in the context of graph homomorphisms, including Oligomorphic Groups and Homogeneous Graphs, with a focus on symmetry and eigenvectors.
Abstract: Preface. Isomorphism and Cayley Graphs on Abelian Groups B. Alspach. Oligomorphic Groups and Homogeneous Graphs P.J. Cameron. Symmetry and Eigenvectors A. Chan, C.D. Godsil. Graph Homomorphisms: Structure and Symmetry G. Hahn, C. Tardif. Cayley Graphs and Interconnection Networks B. Mohar. Finite Transitive Permutation Groups and Finite Vertex-Transitive Graphs C.E. Praeger. Vertex-Transitive Graphs and Digraphs R. Scapellato. Ends and Automorphisms of Infinite Graphs M.E. Watkins. Index.
117 citations
TL;DR: The upper bound for chromatic number and clique covering number of polygon-circle graphs in terms of their clique and independence numbers is of order 2ω, which is better than previously known upper bounds for circle graphs.
113 citations
TL;DR: This work designs optimal, i.e., O(n log n), algorithms for chromatic number, weighted independent set, clique cover and maximum weighted clique on trapezoid graphs and proposes a new class of graphs called circle trapezoidal graphs, which contains trapezoids graphs, circle graphs and circular-arc graphs as subclasses.
107 citations
19 Oct 1997
TL;DR: A polynomial time algorithm is given for the Hamiltonian cycle problem in solid grid graphs, resolving a longstanding open question and identifying Hamiltonian cycles in quad quad graphs, a class of graphs that properly includes solid grid graph.
Abstract: A grid graph is a finite node induced subgraph of the infinite two dimensional integer grid. A solid grid graph is a grid graph without holes. For general grid graphs, the Hamiltonian cycle problem is known to be NP complete. We give a polynomial time algorithm for the Hamiltonian cycle problem in solid grid graphs, resolving a longstanding open question posed by A. Itai et al. (1982). In fact, our algorithm can identify Hamiltonian cycles in quad quad graphs, a class of graphs that properly includes solid grid graphs.
05 Jan 1997
TL;DR: An O(n + m) algorithm for the transitive orientation problem is described, where n and m are the number of vertices and edges of the graph, which gives linear time bounds for maximum clique and minimum vertex coloring on comparability graphs.
Abstract: The transitive orientation problem is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We describe an O(n + m) algorithm for the transitive orientation problem, where n and m are the number of vertices and edges of the graph; full details are given in. This gives linear time bounds for maximum clique and minimum vertex coloring on comparability graphs, recognition of two-dimensional partial orders, permutation graphs, cointerval graphs, and triangulated comparability graphs, and other combinatorial problems on comparability graphs and their complements.
TL;DR: Here propertiesP which do not imply quasi-randomnes for sequences of graphs on their own, but do imply if they hold not only for the whole graph but also for every sufficiently large subgraph of graphs, are investigated.
Abstract: Recently much attention has been focused on the theory of quasi-random graph and hypergraph properties. The class of quasi-random graphs is defined by certain equivalent graph properties possessed by random graphs. We shall investigate propertiesP which do not imply quasi-randomnes for sequences (G
n
) of graphs on their own, but do imply if they hold not only for the whole graphG
n
but also for every sufficiently large subgraph ofG
n
. Here the properties are strongly connected to countingnot necessarily induced subgraphs of a given type, while in a subsequent paper we shall investigate the properties connected with counting induced subgraphs.
TL;DR: The complexity status of the MINIMUM t -spanner problem for various values of t is completely settled and approximation algorithms for the bandwidth minimization problem on convex bipartite graphs and split graphs using the notion of tree spanners are provided.
Abstract: A t -spanner of a graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times their distance in G . Spanners arise in the context of approximating the original graph with a sparse subgraph (Peleg, D., and Schaffer, A. A. (1989), J. Graph. Theory 13 (1), 99–116). The MINIMUM t -SPANNER problem seeks to find a t -spanner with the minimum number of edges for the given graph. In this paper, we completely settle the complexity status of this problem for various values of t , on chordal graphs, split graphs, bipartite graphs and convex bipartite graphs. Our results settle an open question raised by L. Cai (1994, Discrete Appl. Math. 48 , 187–194) and also greatly simplify some of the proofs presented by Cai and by L. Cai and M. Keil (1994, Networks 24 , 233–249). We also give a factor 2 approximation algorithm for the MINIMUM 2-SPANNER problem on interval graphs. Finally, we provide approximation algorithms for the bandwidth minimization problem on convex bipartite graphs and split graphs using the notion of tree spanners.
TL;DR: This paper proves that any LexBFS-ordering of a chordal graph is a common perfect elimination ordering of all odd powers of this graph.
TL;DR: A simple structural characterization of well-covered graphs is presented and a new polynomial time algorithm is presented for the case where the input graph contains no induced subgraph isomorphic to K1, 3.
01 Aug 1997
TL;DR: This work shows how to model persistent graphs in functional languages by graph constructors, and presents a promotion theorem for one of these folds that allows program fusion and the elimination of intermediate results.
Abstract: Graph algorithms expressed in functional languages often suffer from their inherited imperative, state-based style. In particular, this impedes formal program manipulation. We show how to model persistent graphs in functional languages by graph constructors. This provides a decompositional view of graphs which is very close to that of data types and leads to a "more fictional" formulation of graph algorithms. Graph constructors enable the definition of general fold operations for graphs. We present a promotion theorem for one of these folds that allows program fusion and the elimination of intermediate results. Fusion is not restricted to the elimination of tree-like structures, and we prove another theorem that facilitates the elimination of intermediate graphs. We describe an ML-implementation of persistent graphs which efficiently supports the presented fold operators. For example, depth-first-search expressed by a fold over a functional graph has the same complexity as the corresponding imperative algorithm.
TL;DR: Practical algorithms for constructing partitions of graphs into a number of vertex-disjoint subgraphs that satisfy particular degree constraints that are applicable to k-cuts of graphs are presented.
Abstract: We present practical algorithms for constructing partitions of graphs into a xed number of vertex-disjoint subgraphs that satisfy particular degree constraints. We use this in particular to nd k-cuts of graphs of
TL;DR: This paper shows how to compute four measures of vulnerability, i.e., the toughness, scattering number, vertex integrity and the size of a minimum balanced separator by polynomial-time algorithms for various classes of intersection graphs like permutation graphs, bounded dimensional cocomparability graphs, interval graphs, trapezoid graphs and circular versions of these graphs.
TL;DR: A graph isbridged if it contains no isometric cycles of length greater than three and if every vertexvi,i>1, is dominated by some neighbourvj,j.
TL;DR: It is shown that bull-free perfect graphs are quasi-parity graphs, and thatBull- free perfect graphs with no antihole are perfectly contractile.
Abstract: A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle Chvatal and Sbihi showed that the Strong Perfect Graph Conjecture holds for bull-free graphs We show that bull-free perfect graphs are quasi-parity graphs, and that bull-free perfect graphs with no antihole are perfectly contractile Our proof yields a polynomial algorithm for coloring bull-free strict quasi-parity graphs
TL;DR: An efficient algorithm is proposed for solving the clique r-domination problem and the cliques r-packing problem on dually chordal graphs which are a natural generalization of strongly chordalGraphs.
Abstract: Let $\cal C$ be a family of cliques of a graph G=(V,E). Suppose that each clique C of $\cal C$ is associated with an integer r(C)$, where $r(C) \ge 0$. A vertex v r-dominates a clique C of G if $d(v,x) \le r(C)$ for all $x \in C$, where d(v,x) is the standard graph distance. A subset $D \subseteq V$ is a clique r-dominating set of G if for every clique $C \in \cal C$ there is a vertex $u \in D$ which r-dominates C. A clique r-packing set is a subset $P \subseteq \cal C$ such that there are no two distinct cliques $C',C''\in P$ $r$-dominated by a common vertex of G. The clique r-domination problem is to find a clique r-dominating set with minimum size and the clique r-packing problem is to find a clique r-packing set with maximum size. The formulated problems include many domination and clique-transversal-related problems as special cases. In this paper an efficient algorithm is proposed for solving these problems on dually chordal graphs which are a natural generalization of strongly chordal graphs. The efficient algorithm is mainly based on the tree structure and special vertex elimination orderings of dually chordal graphs. In some important particular cases where the algorithm works in linear time the obtained results generalize and improve known results on strongly chordal graphs.
TL;DR: In this paper, the authors consider the problem of coloring perfect graphs with precolored vertices and give a sharp border between the polynomial and NP-complete instances, when precoline vertices occur.
Abstract: We consider the question of the computational complexity of coloring perfect graphs with some precolored vertices. It is well known that a perfect graph can be colored optimally in polynomial time. Our results give a sharp border between the polynomial and NP-complete instances, when precolored vertices occur. The key result on the polynomially solvable cases includes a good characterization theorem on the existence of an optimal coloring of a perfect graph where a given stable set is precolored with only one color. The key negative result states that the 3-colorability of a graph whose odd circuits go through two fixed vertices is NP-complete. The polynomial algorithms use Grotschel, Lovasz and Schrijver's algorithm for finding a maximum clique in a graph, but are otherwise purely combinatorial. c © 1997 John Wiley & Sons, Inc. J Graph Theory 25:
18 Jun 1997
TL;DR: It is shown that there are efficient algorithms to compute the asteroidal number for claw- free graphs, HHD-free graphs, circular-arc graphs and circular permutation graphs, while the corresponding decision problem for graphs in general is NP-complete.
Abstract: A set A of vertices of a graph G = (V, E) is an asteroidal set if for each vertex a ɛ A, the set A {a} is contained in one connected component of G−N[a]. The maximum cardinality of an asteroidal set of the graph G is said to be the asteroidal number of G. We show that there are efficient algorithms to compute the asteroidal number for claw-free graphs, HHD-free graphs, circular-arc graphs and circular permutation graphs, while the corresponding decision problem for graphs in general is NP-complete.
TL;DR: This paper proposes a linear recognition algorithm for semi- P 4 -sparse graphs and shows that with very little work, it can extend the linear algorithms of Chvatal et al. (1987) concerning the class of perfect graphs that are P 5, P 5 and C 5 -free.
TL;DR: This paper shows how to obtain efficient algorithms to compute a minimum cardinality total dominating set on a variety of graph classes, among them permutation graphs, dually chordal graphs and k -polygon graphs.
TL;DR: It is shown that a graph with clique covering number two is a circular arc graph if and only if its edges can be coloured by two colours so that no induced four-cycle contains two opposite edges of the same colour.
Abstract: In our first remark we observe a property of circular arcs which is similar to the Helly property and is helpful in describing all maximal cliques in circular arc graphs (as well as allowing us to genralize a result of Tucker). Our main result is a new simple characterization of circular arc graphs of clique covering number two. These graphs play a crucial role in recognition algorithms for circular arc graphs, and have been characterized by several authors. Specifically, we show that a graph with clique covering number two is a circular arc graph if and only if its edges can be coloured by two colours so that no induced four-cycle contains two opposite edges of the same colour. Our proof of the characterization depends on the ‘lexicographic method’ we have recently introduced. Both remarks could be useful in designing efficient algorithms for (maximum cliques in, respectively recognition of) circular arc graphs
TL;DR: It is shown that the VC-dimension for set systems induced by stars is computable in polynomial time, and the extremal graphs G with the minimum number of edges such that VC P ( G ) ⩾ k .
TL;DR: An infinite set A of finite graphs such that for any graph G e A the order | V(kn(G))| of the n-th iterated clique graph kn(G) is a linear function of n.
Abstract: We present an infinite set A of finite graphs such that for any graph G e A the order | V(k n (G))| of the n-th iterated clique graph k n (G) is a linear function of n. We also give examples of graphs G such that | V(k n(G))| is a polynomial of any given positive degree.
07 Jul 1997
TL;DR: There is an O(n 2 · (¯m+1) time algorithm to compute the maximum cardinality of an independent set for AT-free graphs, and it is observed that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT- free graphs.
Abstract: An asteroidal triple is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an asteroidal triple. We show that there is an O(n 2 · (¯m+1)) time algorithm to compute the maximum cardinality of an independent set for AT-free graphs, where n is the number of vertices and ¯m is the number of non edges of the input graph. Furthermore we obtain O(n 2 · (¯m+1)) time algorithms to solve the INDEPENDENT DOMINATING SET and the INDEPENDENT PERFECT DOMINATING SET problem on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally we observe that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT-free graphs.