Minimally 2-edge connected graphs

A constructive characterization of minimally 2-edge connected graphs, similar to those of Dirac for minimally 2-connected graphs is given.     

Finding 2-edge connected spanning subgraphs

This paper studies the NP-hard problem of finding a minimum size 2-edge connected spanning subgraph (2-ECSS). An algorithm is given that on an r-edge connected input graph G=(V,E) finds a 2-ECSS of size at most |V|+(|E|−|V|)/(r−1). For r-regular, r-edge connected input graphs for r=3, 4, 5 and 6, this gives approximation guarantees of and , respectively.     

Critical extreme points of the 2-edge connected spanning subgraph polytope

In this paper we study the extreme points of the polytope P(G), the linear relaxation of the 2-edge connected spanning subgraph polytope of a graph G. We introduce a partial ordering on the extreme points of P(G) and give necessary conditions for a non-integer extreme point of P(G) to be minimal with respect to that ordering. We show that, if is a non-integer minimal extreme point of P(G), then G and can be reduced, by means of some reduction operations, to a graph G' and an extreme point of P(G') where G' and satisfy some simple properties. As a consequence we obtain a characterization of the perfectly 2-edge connected graphs, the graphs for which the polytope P(G) is integral. Received: May, 2004 Part of this work has been done while the first author was visiting the Research Institute for Discrete Mathematics, University of Bonn, Germany.     

Extensions on 2-edge connected 3-regular up-embeddable graphs

1.IntroductionLetHbea3-regulargraph.Forel,e2,e3EE(H)(el,eZande3areallowedtobethesame),weaddthreenewvenicesal)wZandw3inel,eZande3respectively.ChoosingueV(H),andthenjoiningutofi(i=1,2,3),weatlastobtaina3-regulargraphG.Or,inotherwords,wesaythatGisobtainedfromHbyanM-extension.DenoteG=M(u)(H)(seeFig.1).LetHbea3-regulargraph.Takingel,eZEE(H)(elandeZareallowedtobethesame),weputtwodistinctvenicestvian0dwZinelandeZrespectively,andaddtwodistinctvenicesu,v4V(H),thenjoinutoal,joinvtowZandjoin…     

Every connected graph is a query graph

Let V be a set of bit strings of length k, i.e., V ? {0, 1}^k. The query graph Q(V) is defined as follows: the vertices of Q(V) are the elements of V, and {ū, v?} is an edge of Q(V) if and only if no other w? ? V agrees with ū in all the positions in which v? does. If V represents the set of keys for a statistical data base in which queries that match only one key are rejected, then the security of the individual data is related to the graph Q(V). Ernst Leiss showed that graphs belonging to any of several classes could be represented as query graphs and asked whether any connected graph could be so represented. We answer his question in the affirmative by making use of a spanning tree with special properties.     

Forbidden subgraphs and hamiitonian properties in the square of a connected graph

Various Harniltonian-like properties are investigated in the squares of connected graphs free of some set of forbidden subgraphs. The star K_1,4 the subdivision graph of K_1,3, and the subdivision graph of K_1,3 minus an endvertex play central roles. In particular, we show that connected graphs free of the subdivision graph of K_1,3 minus an endvertex have vertex pancyclic squares.     

On the distance function of a connected graph

An axiomatic characterization of the distance function of a connected graph is given in this note. The triangle inequality is not contained in this characterization.     

Hamiltonian properties of locally connected graphs with bounded vertex degree

We consider the existence of Hamiltonian cycles for the locally connected graphs with a bounded vertex degree. For a graph G, let Δ(G) and δ(G) denote the maximum and minimum vertex degrees, respectively. We explicitly describe all connected, locally connected graphs with Δ(G)?4. We show that every connected, locally connected graph with Δ(G)=5 and δ(G)?3 is fully cycle extendable which extends the results of Kikust [P.B. Kikust, The existence of the Hamiltonian circuit in a regular graph of degree 5, Latvian Math. Annual 16 (1975) 33-38] and Hendry [G.R.T. Hendry, A strengthening of Kikust’s theorem, J. Graph Theory 13 (1989) 257-260] on full cycle extendability of the connected, locally connected graphs with the maximum vertex degree bounded by 5. Furthermore, we prove that problem Hamilton Cycle for the locally connected graphs with Δ(G)?7 is NP-complete.     

The circumference of the square of a connected graph

The celebrated result of Fleischner states that the square of every 2-connected graph is Hamiltonian. We investigate what happens if the graph is just connected. For every n ≥ 3, we determine the smallest length c(n) of a longest cycle in the square of a connected graph of order n and show that c(n) is a logarithmic function in n. Furthermore, for every c ≥ 3, we characterize the connected graphs of largest order whose square contains no cycle of length at least c.     

The entire graph of a bridgeless connected plane graph is hamiltonian

In this paper we show that the entire graph of a bridgeless connected plane graph is hamiltonian, and that the entire graph of a plane block is hamiltonian connected and vertex pancyclic. In addition, we show that in any block G which is not a circuit, given a vertex v of G and a circuit k of G, there is a path p, suspended in G, such that p is a path in k of length at least 1 and G ? E(p) ? V₀(G ? E(p)) is a block which includes v.     

On the doubly connected domination number of a graph

For a given connected graph G = (V, E), a set is a doubly connected dominating set if it is dominating and both 〈D〉 and 〈V (G)-D〉 are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.     

Bounds on the connected domination number of a graph

A subset S$S$ of vertices in a graph G=(V,E)$G=(V,E)$ is a connected dominating set of G$G$ if every vertex of V?S$V?S$ is adjacent to a vertex in S$S$ and the subgraph induced by S$S$ is connected. The minimum cardinality of a connected dominating set of G$G$ is the connected domination number _γc(G)${\gamma }_c\left(G\right)$. The girth g(G)$g\left(G\right)$ is the length of a shortest cycle in G$G$. We show that if G$G$ is a connected graph that contains at least one cycle, then _γc(G)≥g(G)−2${\gamma }_c\left(G\right)\ge g\left(G\right)-2$, and we characterize the graphs obtaining equality in this bound. We also establish various upper bounds on the connected domination number of a graph, as well as Nordhaus–Gaddum type results.     

Every finite strongly connected digraph of stability 2 has a Hamiltonian path

Two circuits C₁ and C₂ in a digraph are called consistent circuits if and only if their intersection is either empty, a singleton or a subpath of both C₁ and C₂. It is proved that Every finite strongly connected digraph of G of stability at most 2 is spanned by two consistent circuits. As a consequence, every finite strongly connected digraph of stability two has a Hamiltonian path.     

The square of a block is Hamiltonian connected

Let B be a block (finite connected graph without cut-vertices) with at least four vertices and ξ, η be distinct vertices of B. We construct a new block M = M(B, ξ, η) containing five copies of B, and use the existence of a Hamiltonian circuit in M² to establish the existence of a Hamiltonian path starting at ξ and ending at η in B². A variant of this trick shows that B² ? ξ has a Hamiltonian circuit.     

The w-median of a connected strongly chordal graph

Suppose G = (V, E) is a graph in which every vertex x has a non-negative real number w(x) as its weight. The w-distance sum of a vertex y is D_{G, w}(y) = σ_x?v d(y, x)w(x). The w-median of G is the set of all vertices y with minimum w-distance sum D_G,w(y). This paper shows that the w-median of a connected strongly chordal graph G is a clique when w(x) is positive for all vertices x in G.     

Every connected regular graph of even degree is a Schreier coset graph

Using Petersen's theorem, that every regular graph of even degree is 2-factorable, it is proved that every connected regular graph of even degree is isomorphic to a Schreier coset graph. The method used is a special application of the permutation voltage graph construction developed by the author and Tucker. This work is related to graph imbedding theory, because a Schreier coset graph is a covering space of a bouquet of circles.     

Enumeration of connected graph coverings

The number of the isomorphism classes of n-fold coverings of a graph G is enumerated by the authors (Canad. J. Math. XLII (1990), 747–761) and Hofmeister (Discrete Math. 98 (1991), 175–183). But the enumeration of the isomorphism classes of connected n-fold coverings of a graph has not been studied except for n = 2. In this paper, we enumerate the isomorphism classes of connected n-fold coverings of a graph G for any n. As a consequence, we obtain a formula for finding the total number of conjugacy classes of subgroups of a given index of a finitely generated free group. © 1996 John Wiley & Sons, Inc.