On Models of Directed Path Graphs Non Rooted Directed Path Graphs

A directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted directed path graph is the intersection graph of a family of directed subpaths of a rooted tree. Clearly, rooted directed path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to find such characterizations for rooted directed path graphs. With the purpose of proving knowledge in this direction, we show in this paper properties of directed path models that can not be rooted for chordal graphs with any leafage and with leafage four. Therefore, we prove that for leafage four directed path graphs minimally non rooted directed path graphs have a unique asteroidal quadruple, and can be characterized by the presence of certain type of asteroidal quadruples.     

图的路色数问题的NP－完全性

一个给定的图是否存在用ｒ种颜色的正常Ｐｋ着色？称该问题为图的（ｋ，ｒ）路色数问题．本文研究其算法复杂性，并得到以下结果：对于任意给定的ｋ，２≤ｋ≤∞，图的（ｋ，２）路色数问题及直径为２的图的（ｋ，３）路色数问题都是ＮＰ－完全的；对于任意给定的ｋ，２≤ｋ≤∞，平面图的（ｋ，３）路色数问题也是ＮＰ－完全的．     

An Extremal Problem For Random Graphs And The Number Of Graphs With Large Even-Girth

C _{2 k} -free subgraph of a random graph may have, obtaining best possible results for a range of p=p(n). Our estimates strengthen previous bounds of Füredi [12] and Haxell, Kohayakawa, and Łuczak [13]. Two main tools are used here: the first one is an upper bound for the number of graphs with large even-girth, i.e., graphs without short even cycles, with a given number of vertices and edges, and satisfying a certain additional pseudorandom condition; the second tool is the powerful result of Ajtai, Komlós, Pintz, Spencer, and Szemerédi [1] on uncrowded hypergraphs as given by Duke, Lefmann, and R?dl [7]. Received: February 17, 1995     

Word Problem of the Perkins Semigroup via Directed Acyclic Graphs

For a word w in an alphabet Γ, the alternation word digraph Alt(w), a certain directed acyclic graph associated with w, is presented as a means to analyze the free spectrum of the Perkins monoid \(\mathbf{B_2^1}\). Let \((f_n^{\mathbf{B_2^1}})\) denote the free spectrum of \(\mathbf{B_2^1}\), let a _n be the number of distinct alternation word digraphs on words whose alphabet is contained in {x ₁,..., x _n }, and let p _n denote the number of distinct labeled posets on {1,..., n}. The word problem for the Perkins semigroup \(\mathbf{B_2^1}\) is solved here in terms of alternation word digraphs: Roughly speaking, two words u and v are equivalent over \(\mathbf{B_2^1}\) if and only if certain alternation graphs associated with u and v are equal. This solution provides the main application, the bounds: \(p_n \leq a_n \leq f_n^{\mathbf{B_2^1}} \leq 2^{n}a_{2n}^2\). A result of the second author in a companion paper states that \((\operatorname{log} \; a_n)\in O(n^3)\), from which it follows that \((\operatorname{log} f_n^{\mathbf{B_2^1}})\in O(n^3)\) as well. Alternation word digraphs are of independent interest combinatorially. It is shown here that the computational complexity problem that has as instance {u,v} where u,v are words of finite length, and question “Is Alt(u)?=?Alt(v)?”, is co-NP-complete. Additionally, alternation word digraphs are acyclic, and certain of them are natural extensions of posets; each realizer of a finite poset determines an extension by an alternation word digraph.     

Complement and Isomorphism on Bipolar Fuzzy Graphs

In this paper, we discuss some properties of the self complement and self weak complement bipolar fuzzy graphs, and get a sufficient condition for a bipolar fuzzy graph to be the self weak complement bipolar fuzzy graph. Also we investigate relations between operations union, join, and complement on bipolar fuzzy graphs.     

The Isomorphism Problem for All Hyperbolic Groups

We give a solution to Dehn’s isomorphism problem for the class of all hyperbolic groups, possibly with torsion. We also prove a relative version for groups with peripheral structures. As a corollary, we give a uniform solution to Whitehead’s problem asking whether two tuples of elements of a hyperbolic group G are in the same orbit under the action of Aut(G). We also get an algorithm computing a generating set of the group of automorphisms of a hyperbolic group preserving a peripheral structure.     

The Singularity/Nonsingularity Problem for Matrices Satisfying Diagonal Dominance Conditions in Terms of Directed Graphs

The paper considers the singularity/nonsingularity problem for matrices satisfying certain conditions of diagonal dominance. The conditions considered extend the classical diagonal dominance conditions and involve the directed graph of the matrix in question. Furthermore, in the case of the so-called mixed diagonal dominance, the corresponding conditions are allowed to involve both row and column sums for an arbitrary finite set of matrices diagonally conjugated to the original matrix. Conditions sufficient for the nonsingularity of quasi-irreducible matrices strictly diagonally dominant in certain senses are established, as well as necessary and sufficient conditions of singularity/nonsingularity for weakly diagonally dominant matrices in the irreducible case. The results obtained are used to describe inclusion regions for eigenvalues of arbitrary matrices. In particular, a direct extension of the Gerschgorin (r = 1) and Ostrowski-Brauer (r = 2) theorems to r ≥ 3 is presented. Bibliography: 18 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 40–83.     

Small Directed Strongly Regular Graphs

The goal of the present paper is to provide a gallery of small directed strongly regular graphs.For each graph of order n≤12 and valency k相似文献   

Operator Machines on Directed Graphs

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator ${R\,:\,X\longrightarrow\, X}We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X? X{R\,:\,X\longrightarrow\, X} such that the set

A = {x ? X : ||Rn x||? ￥}A = \{x \in X\,:\,{\left|\left|{R^n x}\right|\right|}\rightarrow \infty\}      12. Chip-Firing Games on Directed Graphs      Anders Björner  László Lovász 《Journal of Algebraic Combinatorics》1992,1(4):305-328 We consider the following (solitary) game: each node of a directed graph contains a pile of chips. A move consists of selecting a node with at least as many chips as its outdegree, and sending one chip along each outgoing edge to its neighbors. We extend to directed graphs several results on the undirected version obtained earlier by the authors, P. Shor, and G. Tardos, and we discuss some new topics such as periodicity, reachability, and probabilistic aspects.Among the new results specifically concerning digraphs, we relate the length of the shortest period of an infinite game to the length of the longest terminating game, and also to the access time of random walks on the same graph. These questions involve a study of the Laplace operator for directed graphs. We show that for many graphs, in particular for undirected graphs, the problem whether a given position of the chips can be reached from the initial position is polynomial time solvable.Finally, we show how the basic properties of the probabilistic abacus can be derived from our results.      13. Minimum Paths in Directed Graphs      A. Frieze 《The Journal of the Operational Research Society》1977,28(2):339-346 This paper considers the problem of finding paths from a fixed node to all other nodes of a directed graph which minimise a function defined on the paths. Under certain assumptions a characterisation of optimal paths is derived. Two algorithms which are generalisations of standard shortest path methods are then given.      14. Even Cycles in Directed Graphs      Carsten Thomassen 《European Journal of Combinatorics》1985,6(1):85-89       15. List-Compactness of Infinite Directed Graphs      Bruce L. Bauslaugh 《Graphs and Combinatorics》2001,17(1):17-38 A digraph H is homomorphically compact if the digraphs G which admit homomorphisms to H are exactly the digraphs whose finite subdigraphs all admit homomorphisms to H. In this paper we define a similar notion of compactness for list-homomorphisms. We begin by showing that it is essentially only finite digraphs that are compact with respect to list-homomorphisms. We then explore the effects of restricting the types of list-assignments which are permitted, and obtain some richer characterizations. Received: May 16, 1997 Final version received: January 16, 1998      16. The Ferry Cover Problem on Regular Graphs and Small-Degree Graphs      Erfang SHAN  Liying KANG 《数学年刊B辑(英文版)》2018,39(6):933-946 The ferry problem may be viewed as generalizations of the classical wolf-goat-cabbage puzzle. The ferry cover problem is to determine the minimum required boat capacity to safely transport n items represented by a conflict graph. The Alcuin number of a conflict graph is the smallest capacity of a boat for which the graph possesses a feasible ferry schedule. In this paper the authors determine the Alcuin number of regular graphs and graphs with maximum degree at most five.      17. On Rooted Packings,Decompositions, and Factors of Graphs          下载免费PDF全文 Zbigniew Lonc  Nataliya Petryshyn 《Journal of Graph Theory》2014,76(2):106-128 In this article, we study so‐called rooted packings of rooted graphs. This concept is a mutual generalization of the concepts of a vertex packing and an edge packing of a graph. A rooted graph is a pair , where G is a graph and . Two rooted graphs and are isomorphic if there is an isomorphism of the graphs G and H such that S is the image of T in this isomorphism. A rooted graph is a rooted subgraph of a rooted graph if H is a subgraph of G and . By a rooted ‐packing into a rooted graph we mean a collection of rooted subgraphs of isomorphic to such that the sets of edges are pairwise disjoint and the sets are pairwise disjoint. In this article, we concentrate on studying maximum ‐packings when H is a star. We give a complete classification with respect to the computational complexity status of the problems of finding a maximum ‐packing of a rooted graph when H is a star. The most interesting polynomial case is the case when H is the 2‐edge star and S contains the center of the star only. We prove a min–max theorem for ‐packings in this case.      18. Regular Subgraphs in Graphs and Rooted Graphs and Definability in Monadic Second - Order Logic      Iain A. Stewart 《Mathematical Logic Quarterly》1997,43(1):1-21 We investigate the definability in monadic ∑11 and monadic Π11 of the problems REGk, of whether there is a regular subgraph of degree k in some given graph, and XREGk, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REGkk and XREGkk, respectively, for k ≥ 2 (all our graphs are undirected). Our motivation partly stems from the fact (which we prove here) that REGkk and XREGkk are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first - order reductions, monadic ∑11 games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑11 and monadic Π11, and we compare the definability of these problems (in monadic ∑11 and monadic Π11 with their computational complexity (which varies from solvable using logspace to NP - complete).      19. The Ferry Cover Problem on Regular Graphs and Small-Degree GraphsThe Ferry Cover Problem on Regular Graphs and Small-Degree Graphs      《数学年刊B辑(英文版)》2018,(6) The ferry problem may be viewed as generalizations of the classical wolf-goatcabbage puzzle. The ferry cover problem is to determine the minimum required boat capacity to safely transport n items represented by a conflict graph. The Alcuin number of a conflict graph is the smallest capacity of a boat for which the graph possesses a feasible ferry schedule. In this paper the authors determine the Alcuin number of regular graphs and graphs with maximum degree at most five.      20. First-Order Complexity of Subgraph Isomorphism via Kneser Graphs      Voronov  V. A.  Dergachev  E. A.  Zhukovskii  M. E.  Neopryatnaya  A. M. 《Mathematical Notes》2021,109(1-2):29-37 Mathematical Notes - The problem of finding the least number of variables of a first-order formula expressing the statement that an input-graph contains a subgraph isomorphic to a given...

Chip-Firing Games on Directed Graphs

We consider the following (solitary) game: each node of a directed graph contains a pile of chips. A move consists of selecting a node with at least as many chips as its outdegree, and sending one chip along each outgoing edge to its neighbors. We extend to directed graphs several results on the undirected version obtained earlier by the authors, P. Shor, and G. Tardos, and we discuss some new topics such as periodicity, reachability, and probabilistic aspects.Among the new results specifically concerning digraphs, we relate the length of the shortest period of an infinite game to the length of the longest terminating game, and also to the access time of random walks on the same graph. These questions involve a study of the Laplace operator for directed graphs. We show that for many graphs, in particular for undirected graphs, the problem whether a given position of the chips can be reached from the initial position is polynomial time solvable.Finally, we show how the basic properties of the probabilistic abacus can be derived from our results.     

Minimum Paths in Directed Graphs

This paper considers the problem of finding paths from a fixed node to all other nodes of a directed graph which minimise a function defined on the paths. Under certain assumptions a characterisation of optimal paths is derived. Two algorithms which are generalisations of standard shortest path methods are then given.     

List-Compactness of Infinite Directed Graphs

A digraph H is homomorphically compact if the digraphs G which admit homomorphisms to H are exactly the digraphs whose finite subdigraphs all admit homomorphisms to H. In this paper we define a similar notion of compactness for list-homomorphisms. We begin by showing that it is essentially only finite digraphs that are compact with respect to list-homomorphisms. We then explore the effects of restricting the types of list-assignments which are permitted, and obtain some richer characterizations. Received: May 16, 1997 Final version received: January 16, 1998     

The Ferry Cover Problem on Regular Graphs and Small-Degree Graphs

The ferry problem may be viewed as generalizations of the classical wolf-goat-cabbage puzzle. The ferry cover problem is to determine the minimum required boat capacity to safely transport n items represented by a conflict graph. The Alcuin number of a conflict graph is the smallest capacity of a boat for which the graph possesses a feasible ferry schedule. In this paper the authors determine the Alcuin number of regular graphs and graphs with maximum degree at most five.     

On Rooted Packings,Decompositions, and Factors of Graphs

In this article, we study so‐called rooted packings of rooted graphs. This concept is a mutual generalization of the concepts of a vertex packing and an edge packing of a graph. A rooted graph is a pair , where G is a graph and . Two rooted graphs and are isomorphic if there is an isomorphism of the graphs G and H such that S is the image of T in this isomorphism. A rooted graph is a rooted subgraph of a rooted graph if H is a subgraph of G and . By a rooted ‐packing into a rooted graph we mean a collection of rooted subgraphs of isomorphic to such that the sets of edges are pairwise disjoint and the sets are pairwise disjoint. In this article, we concentrate on studying maximum ‐packings when H is a star. We give a complete classification with respect to the computational complexity status of the problems of finding a maximum ‐packing of a rooted graph when H is a star. The most interesting polynomial case is the case when H is the 2‐edge star and S contains the center of the star only. We prove a min–max theorem for ‐packings in this case.     

Regular Subgraphs in Graphs and Rooted Graphs and Definability in Monadic Second - Order Logic

We investigate the definability in monadic ∑₁¹ and monadic Π₁¹ of the problems REG_k, of whether there is a regular subgraph of degree k in some given graph, and XREG_k, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REG_k^k and XREG_k^k, respectively, for k ≥ 2 (all our graphs are undirected). Our motivation partly stems from the fact (which we prove here) that REG_k^k and XREG_k^k are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first - order reductions, monadic ∑₁¹ games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑₁¹ and monadic Π₁¹, and we compare the definability of these problems (in monadic ∑₁¹ and monadic Π₁¹ with their computational complexity (which varies from solvable using logspace to NP - complete).     

The Ferry Cover Problem on Regular Graphs and Small-Degree GraphsThe Ferry Cover Problem on Regular Graphs and Small-Degree Graphs

The ferry problem may be viewed as generalizations of the classical wolf-goatcabbage puzzle. The ferry cover problem is to determine the minimum required boat capacity to safely transport n items represented by a conflict graph. The Alcuin number of a conflict graph is the smallest capacity of a boat for which the graph possesses a feasible ferry schedule. In this paper the authors determine the Alcuin number of regular graphs and graphs with maximum degree at most five.     

First-Order Complexity of Subgraph Isomorphism via Kneser Graphs

Mathematical Notes - The problem of finding the least number of variables of a first-order formula expressing the statement that an input-graph contains a subgraph isomorphic to a given...