Eigenvalues of graphs and digraphs

We show that given a digraph X there is a vertex-transitive digraph Y such that the minimal polynomial of X divides that of Y. Two consequences of this are that there exist vertex-transitive digraphs with nondiagonalizable adjacency matrices and that any algebraic integer is the eigenvalue of some vertex-transitive digraph.     

Distance connectivity in graphs and digraphs

Let G = (V, A) be a digraph with diameter D ≠ 1. For a given integer 2 ≤ t ≤ D, the t-distance connectivity κ(t) of G is the minimum cardinality of an x → y separating set over all the pairs of vertices x, y which are at distance d(x, y) ≥ t. The t-distance edge connectivity λ(t) of G is defined similarly. The t-degree of G, δ(t), is the minimum among the out-degrees and in-degrees of all vertices with (out- or -in-) eccentricity at least t. A digraph is said to be maximally distance connected if κ(t) = δ(t) for all values of t. In this paper we give a construction of a digraph having D − 1 positive arbitrary integers c₂ ≤ … ≤ c_D, D > 3, as the values of its t-distance connectivities κ(2) = c₂, …, κ(D) = c_D. Besides, a digraph that shows the independence of the parameters κ(t), λ(t), and δ(t) is constructed. Also we derive some results on the distance connectivities of digraphs, as well as sufficient conditions for a digraph to be maximally distance connected. Similar results for (undirected) graphs are presented. © 1996 John Wiley & Sons, Inc.     

Local-edge-connectivity in digraphs and oriented graphs

A digraph without any cycle of length two is called an oriented graph. The local-edge-connectivityλ(u,v) of two vertices u and v in a digraph or graph D is the maximum number of edge-disjoint u-v paths in D, and the edge-connectivity of D is defined as . Clearly, λ(u,v)?min{d⁺(u),d^-(v)} for all pairs u and v of vertices in D. Let δ(D) be the minimum degree of D. We call a graph or digraph D maximally edge-connected when λ(D)=δ(D) and maximally local-edge-connected when

λ(u,v)=min{d⁺(u),d^-(v)}     

Cycle extendability in graphs and digraphs

In 1990, Hendry conjectured that all chordal Hamiltonian graphs are cycle extendable, that is, the vertices of each non-Hamiltonian cycle are contained in a cycle of length one greater. Let A be a symmetric (0,1)-matrix with zero main diagonal such that A is the adjacency matrix of a chordal Hamiltonian graph. Hendry’s conjecture in this case is that every k×k principle submatrix of A that dominates a full cycle permutation k×k matrix is a principle submatrix of a (k+1)×(k+1) principle submatrix of A that dominates a (k+1)×(k+1) full cycle permutation matrix. This article generalizes the concept of cycle-extendability to S-extendable; that is, with S⊆{1,2,…,n} and G a graph on n vertices, G is S-extendable if the vertices of every non-Hamiltonian cycle are contained in a cycle length i greater, where i∈S. We investigate this concept in directed graphs and in particular tournaments, i.e., anti-symmetric matrices with zero main diagonal.     

On unavoidable digraphs in orientations of graphs

Let G be a graph, and let D be a directed graph. Write G → D to mean that, no matter how the edges of G are given orientations, a copy of D must appear as a subgraph of the resulting oriented graph. It is proved that among all G for which G → D, the minimum chromatic number is equal to the minimum k for which K_k → hom(D), where hom(D) is the set of homomorphs of D. Next, necessary and sufficient conditions are given for a directed graph to have a homomorphism into a given transitive tournament, directed path, or directed cycle. These results are then applied to various cases of the above theorem. In particular, the minimum chromatic number is evaluated whenever D is an oriented forest, and all D are characterized for which the minimum chromatic number is no more than three.     

Subgraph distances in graphs defined by edge transfers

For two edge-induced subgraphs F and H of the same size in a graph G, the subgraph H can be obtained from F by an edge jump if there exist four distinct vertices u, v, w, and x in G such that uv ε E(F), wx ε E(G) - E(F), and H = F - uv + wx. The subgraph F is j-transformed into H if H can be obtained from F by a sequence of edge jumps. Necessary and sufficient conditions are presented for a graph G to have the property that every edge-induced subgraph of a fixed size in G can be j-transformed into every other edge-induced subgraph of that size. The minimum number of edge jumps required to transform one subgraph into another is called the jump distance. This distance is a metric and can be modeled by a graph. The jump graph J(G) of a graph G is defined as that graph whose vertices are the edges of G and where two vertices of J(G) are adjacent if and only if the corresponding edges of G are independent. For a given graph G, we consider the sequence {{J^k(G)}} of iterated jump graphs and classify each graph as having a convergent, divergent, or terminating sequence.     

Isomorphism of circulant graphs and digraphs

Let S? {1, …, n?1} satisfy ?S = S mod n. The circulant graph G(n, S) with vertex set {v₀, v₁,…, v_n?1} and edge set E satisfies v_iv_j?E if and only if j ? i ∈ S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = ? S. Ádám conjectured that $\text{G(n, S) ? G(n, S′)}$ if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p², and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p² where p is an odd prime, or n is divisible by 2⁴.     

Approximating Euclidean distances by small degree graphs

Given an undirected edge-weighted graphG=(V,E), a subgraphG′=(V,E′) is at-spanner ofG if, for everyu, v∈V, the weighted distance betweenu andv inG′ is at mostt times the weighted distance betweenu andv inG. We consider the problem of approximating the distances among points of a Euclidean metric space: given a finite setV of points in ℝd, we want to construct a sparset-spanner of the complete weighted graph induced byV. The weight of an edge in these graphs is the Euclidean distance between the endpoints of the edge. We show by a simple greedy argument that, for anyt>1 and anyV ⊂ ℝd, at-spannerG ofV exists such thatG has degree bounded by a function ofd andt. The analysis of our bounded degree spanners improves over previously known upper bounds on the minimum number of edges of Euclideant-spanners, even compared with spanners of boundedaverage degree. Our results answer two open problems, one proposed by Vaidya and the other by Keil and Gutwin. The main result of the paper concerns the case of dimensiond=2. It is fairly easy to see that, for somet (t≥7.6),t-spanners of maximum degree 6 exist for any set of points in the Euclidean plane, but it was not known that degree 5 would suffice. We prove that for some (fixed)t, t-spanners of degree 5 exist for any set of points in the plane. We do not know if 5 is the best possible upper bound on the degree. This research was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnológico, Proc 203039/87.4 (Brazil).     

Fundamental groupoids of digraphs and graphs

We introduce the notion of fundamental groupoid of a digraph and prove its basic properties. In particular, we obtain a product theorem and an analogue of the Van Kampen theorem. Considering the category of (undirected) graphs as the full subcategory of digraphs, we transfer the results to the category of graphs. As a corollary we obtain the corresponding results for the fundamental groups of digraphs and graphs. We give an application to graph coloring.     

Characterization of digraphs with equal domination graphs and underlying graphs

A domination graph of a digraph D, dom(D), is created using the vertex set of D and edge {u,v}∈E[dom(D)] whenever (u,z)∈A(D) or (v,z)∈A(D) for every other vertex z∈V(D). The underlying graph of a digraph D, UG(D), is the graph for which D is a biorientation. We completely characterize digraphs whose underlying graphs are identical to their domination graphs, UG(D)=dom(D). The maximum and minimum number of single arcs in these digraphs, and their characteristics, is given.     

The hamiltonicity on the competition graphs of round digraphs

Given a digraph D =(V, A), the competition graph G of D, denoted by C(D), has the same set of vertices as D and an edge between vertices x and y if and only if N_D~+(x)∩N_D~+(y)≠Ф. In this paper, we investigate the competition graphs of round digraphs and give a necessary and sufficient condition for these graphs to be hamiltonian.     

The geodetic numbers of graphs and digraphs

For every two vertices u and v in a graph G,a u-v geodesic is a shortest path between u and v.Let I(u,v)denote the set of all vertices lying on a u-v geodesic.For a vertex subset S,let I(S) denote the union of all I(u,v)for u,v∈S.The geodetic number g(G)of a graph G is the minimum cardinality of a set S with I(S)=V(G).For a digraph D,there is analogous terminology for the geodetic number g(D).The geodetic spectrum of a graph G,denoted by S(G),is the set of geodetic numbers of all orientations of graph G.The lower geodetic number is g~-(G)=minS(G)and the upper geodetic number is g~ (G)=maxS(G).The main purpose of this paper is to study the relations among g(G),g~-(G)and g~ (G)for connected graphs G.In addition,a sufficient and necessary condition for the equality of g(G)and g(G×K_2)is presented,which improves a result of Chartrand,Harary and Zhang.     

The connectivity of large digraphs and graphs

This paper studies the relation between the connectivity and other parameters of a digraph (or graph), namely its order n, minimum degree δ, maximum degree Δ, diameter D, and a new parameter l_pi;, 0 ≤ π ≤ δ ? 2, related with the number of short paths (in the case of graphs l₀ = ?(g ? 1)/2? where g stands for the girth). For instance, let G = (V,A) be a digraph on n vertices with maximum degree Δ and diameter D, so that n ≤ n(Δ, D) = 1 + Δ + Δ ² + … + Δ^D (Moore bound). As the main results it is shown that, if κ and λ denote respectively the connectivity and arc-connectivity of G, . Analogous results hold for graphs. © 1993 John Wiley & Sons, Inc.     

Disjoint sub(di)graphs in digraphs

We survey results and open problems concerning the existence of (arc-)disjoint subdigraphs with certain properties in a digraph. Examples are arc-disjoint spanning strong subdigraphs and disjoint paths with prescribed endvertices.     

The strong metric dimension of graphs and digraphs

Let G be a connected (di)graph. A vertex w is said to strongly resolve a pair u,v of vertices of G if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W. The smallest cardinality of a strong resolving set for G is called the strong dimension of G. It is shown that the problem of finding the strong dimension of a connected graph can be transformed to the problem of finding the vertex covering number of a graph. Moreover, it is shown that computing this invariant is NP-hard. Related invariants for directed graphs are defined and studied.