Partitions and homomorphisms in directed and undirected graphs

Boyle has given a condition for defining a homomorphism in terms of minimal paths for undirected graphs. The purpose of such homomorphisms is to provide a simpler graph which will reflect the structure of the more complex graph, and thereby enable the researcher to make observations which may have been shrouded by a preponderance of nodes and edges. This paper develops Boyle's ideas and introduces further homomorphisms for directed as well as undirected graphs. The relationships between the various homomorphisms are also examined.     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

Nomadic decompositions of bidirected complete graphs

We use to denote the bidirected complete graph on n vertices. A nomadic Hamiltonian decomposition of is a Hamiltonian decomposition, with the additional property that “nomads” walk along the Hamiltonian cycles (moving one vertex per time step) without colliding. A nomadic near-Hamiltonian decomposition is defined similarly, except that the cycles in the decomposition have length n-1, rather than length n. Bondy asked whether these decompositions of exist for all n. We show that admits a nomadic near-Hamiltonian decomposition when .     

The degree sequences of self-complementary graphs

Known necessary conditions for realization of a sequence of integers as the degrees of a self-complementary graph are shown to be sufficient. An algorithm for constructing a realization of such a sequence as degrees of such a graph is illustrated by an example.     

On the representability of totally unimodular matrices on bidirected graphs

We present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding node-edge incidence matrices Q and S such that QB=S. Seymour’s famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called k-sums starting from network matrices and their transposes and two compact representation matrices B₁,B₂ of a certain ten element matroid. Given that B₁,B₂ are binet matrices we examine the k-sums of network and binet matrices. It is shown that thek-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k=2,3. A new class of matrices is introduced, the so-called tour matrices, which generalise network, binet and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under k-sums, as well as under pivoting and other elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the k-sum operation and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any TU matrix.     

On weighted directed graphs

The study of a mixed graph and its Laplacian matrix have gained quite a bit of interest among the researchers. Mixed graphs are very important for the study of graph theory as they provide a setup where one can have directed and undirected edges in the graph. In this article we present a more general structure, namely the weighted directed graphs and supply appropriate generalizations of several existing results for mixed graphs related to singularity of the corresponding Laplacian matrix. We also prove many new combinatorial results relating the Laplacian matrix and the graph structure.     

Free multiflows in bidirected and skew-symmetric graphs

A graph (digraph) G=(V,E) with a set T⊆V of terminals is called inner Eulerian if each nonterminal node v has even degree (resp. the numbers of edges entering and leaving v are equal). Cherkassky and Lovász, independently, showed that the maximum number of pairwise edge-disjoint T-paths in an inner Eulerian graph G is equal to , where λ(s) is the minimum number of edges whose removal disconnects s and T-{s}. A similar relation for inner Eulerian digraphs was established by Lomonosov. Considering undirected and directed networks with “inner Eulerian” edge capacities, Ibaraki, Karzanov, and Nagamochi showed that the problem of finding a maximum integer multiflow (where partial flows connect arbitrary pairs of distinct terminals) is reduced to O(logT) maximum flow computations and to a number of flow decompositions.In this paper we extend the above max-min relation to inner Eulerian bidirected graphs and inner Eulerian skew-symmetric graphs and develop an algorithm of complexity for the corresponding capacitated cases. In particular, this improves the best known bound for digraphs. Our algorithm uses a fast procedure for decomposing a flow with O(1) sources and sinks in a digraph into the sum of one-source-one-sink flows.     

The degree ratio ranking method for directed graphs

One of the most famous ranking methods for digraphs is the ranking by Copeland score. The Copeland score of a node in a digraph is the difference between its outdegree (i.e. its number of outgoing arcs) and its indegree (i.e. its number of ingoing arcs). In the ranking by Copeland score, a node is ranked higher, the higher is its Copeland score. In this paper, we deal with an alternative method to rank nodes according to their out- and indegree, namely ranking the nodes according to their degree ratio, i.e. the outdegree divided by the indegree. To avoid dividing by zero, we add 1 to both the out- as well as indegree of every node. We provide an axiomatization of the ranking by degree ratio using a clone property, which says that the entrance of a clone or a copy (i.e. a node that is in some sense similar to the original node) does not change the ranking among the original nodes. We also provide a new axiomatization of the ranking by Copeland score using the same axioms except that this method satisfies a different clone property. Finally, we modify the ranking by degree ratio by taking only the out- and indegree, but by definition assume nodes with indegree zero to be ranked higher than nodes with positive indegree. We provide an axiomatization of this ranking method using yet another clone property and a maximal property. In this way, we can compare the three ranking methods by their clone property.     

On P-quasigroups and decompositions of complete undirected graphs

In [2], A. Kotzig has introduced the concepts of P-groupoid and P-quasigroup and has shown how these concepts are related to the decomposition of a complete undirected graph into disjoint closed paths. To each closed path of the graph associated with a given P-quasigroup Q there corresponds a cyclic partial transversal in the Latin square L which is defined by the multiplication table of Q. In this paper, it is shown that cyclic transversals are connected with Hamiltonian decompositions of complete undirected graphs having an even number of vertices and a connection between the order of a particular type of P-quasigroup and the length of its cyclic partial transversals is established. An indirect connection with the work of Yap [4] is established via the concept of isotopy.     

On low bound of degree sequences of spanning trees in K-edge-connected graphs

A graph G = (V, E) is k-edge-connected if for any subset E′ ⊆ E,|E′| < k, G − E′ is connected. A d_k-tree T of a connected graph G = (V, E) is a spanning tree satisfying that ∀v ∈ V, d_T(v) ≤ + α, where [·] is a lower integer form and α depends on k. We show that every k-edge-connected graph with k ≥ 2, has a d_k-tree, and α = 1 for k = 2, α = 2 for k ≥ 3. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 87–95, 1998     

Which cospectral graphs have same degree sequences

Two graphs are said to be $L$-cospectral (respectively, $Q$-cospectral) if they have the same (respectively, signless) Laplacian spectra, and a graph $G$ is said to be $L?DS$ (respectively, $Q?DS$) if there does not exist other non-isomorphic graph $H$ such that $H$ and $G$ are $L$-cospectral (respectively, $Q$-cospectral). Let $d_1\left(G\right)\ge d_2\left(G\right)\ge ?\ge d_n\left(G\right)$ be the degree sequence of a graph $G$ with $n$ vertices. In this paper, we prove that except for two exceptions (respectively, the graphs with $d_1\left(G\right)\in \{4,5\}$), if $H$ is $L$-cospectral (respectively, $Q$-cospectral) with a connected graph $G$ and $d_2\left(G\right)=2$, then $H$ has the same degree sequence as $G$. A spider graph is a unicyclic graph obtained by attaching some paths to a common vertex of the cycle. As an application of our result, we show that every spider graph and its complement graph are both $L?DS$, which extends the corresponding results of Haemers et al. (2008), Liu et al. (2011), Zhang et al. (2009) and Yu et al. (2014).     

On the X-join decomposition for undirected graphs

In his paper [17], Sabidussi defined the X-join of a family of graphs. Cowan, James, Stanton gave in [6] and O(n⁴) algorithm that decomposes a graph, when possible, into the X-join of the family of its subgraphs. We give here another approach using an equivalence relation on the edge set of the graph. We prove that if G and its complement are connected then there exists an unique class of edges that covers all the vertices of G. This theorem yields immediately an O(n³) decomposition algorithm.     

Nowhere-zero integral chains and flows in bidirected graphs

General results on nowhere-zero integral chain groups are proved and then specialized to the case of flows in bidirected graphs. For instance, it is proved that every 4-connected (resp. 3-connected and balanced triangle free) bidirected graph which has at least an unbalanced circuit and a nowhere-zero flow can be provided with a nowhere-zero integral flow with absolute values less than 18 (resp. 30). This improves, for these classes of graphs, Bouchet's 216-flow theorem (J. Combin. Theory Ser. B 34 (1982), 279–292). We also approach his 6-flow conjecture by proving it for a class of 3-connected graphs. Our method is inspired by Seymour's proof of the 6-flow theorem (J. Combin. Theory Ser. B 30 (1981), 130–136), and makes use of new connectedness properties of signed graphs.     

On 3-restricted edge connectivity of undirected binary Kautz graphs

An edge cut of a connected graph is m-restricted if its removal leaves every component having order at least m. The size of minimum m-restricted edge cuts of a graph G is called its m-restricted edge connectivity. It is known that when m≤4, networks with maximal m-restricted edge connectivity are most locally reliable. The undirected binary Kautz graph UK(2,n) is proved to be maximal 2- and 3-restricted edge connected when n≥3 in this work. Furthermore, every minimum 2-restricted edge cut disconnects this graph into two components, one of which being an isolated edge.     

On Riesz and Wishart distributions associated with decomposable undirected graphs

Classical Wishart distributions on the open convex cone of positive definite matrices and their fundamental features are extended to generalized Riesz and Wishart distributions associated with decomposable undirected graphs using the basic theory of exponential families. The families of these distributions are parameterized by their expectations/natural parameter and multivariate shape parameter and have a non-trivial overlap with the generalized Wishart distributions defined in Andersson and Wojnar (2004) [4] and [8]. This work also extends the Wishart distributions of type I in Letac and Massam (2007) [7] and, more importantly, presents an alternative point of view on the latter paper.     

A simple criterion on degree sequences of graphs

A finite sequence of nonnegative integers is called graphic if the terms in the sequence can be realized as the degrees of vertices of a finite simple graph. We present two new characterizations of graphic sequences. The first of these is similar to a result of Havel-Hakimi, and the second equivalent to a result of Erd?s & Gallai, thus providing a short proof of the latter result. We also show how some known results concerning degree sets and degree sequences follow from our results.