Removable edges in cycles of a <Emphasis Type="Italic">k</Emphasis>-connected graph

An edge e of a k-connected graph G is said to be a removable edge if G ⊖ e is still k-connected, where G ⊖ e denotes the graph obtained from G by deleting e to get G − e, and for any end vertex of e with degree k − 1 in G − e, say x, delete x, and then add edges between any pair of non-adjacent vertices in N _G−e (x). The existence of removable edges of k-connected graphs and some properties of 3-connected graphs and 4-connected graphs have been investigated. In the present paper, we investigate some properties of k-connected graphs and study the distribution of removable edges on a cycle in a k-connected graph (k ≥ 4).     

Removable edges of cycles in 5-connected graphs

Let G be a 5-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G?e; second, for each vertex x of degree 4 in G?e, delete x from G?e and then completely connect the 4 neighbors of x by K ₄. If multiple edges occur, we use single edge to replace them. The final resultant graph is denoted by G ? e. If G ? e is still 5-connected, then e is called a removable edge of G. In this paper, we investigate the distribution of removable edges in a cycle of a 5-connected graph. And we give examples to show some of our results are best possible in some sense.     

5连通图中的可去边

An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 in G- e, say x, delete x, and then add edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.     

Removable edges in 3-connected graphs

An edge e of a 3-connected graph G is said to be removable if G - e is a subdivision of a 3-connected graph. If e is not removable, then e is said to be nonremovable. In this paper, we study the distribution of removable edges in 3-connected graphs and prove that a 3-connected graph of order n ≥ 5 has at most [(4 n — 5)/3] nonremovable edges.     

The 3-Connected Graphs with Exactly Three Non-Essential Edges

An edge e of a simple 3-connected graph G is essential if neither the deletion G\e nor the contraction G/e is both simple and 3-connected. Tuttes Wheels Theorem proves that the only simple 3-connected graphs with no non-essential edges are the wheels. In earlier work, as a corollary of a matroid result, the authors determined all simple3-connected graphs with at most two non-essential edges. This paper specifies all such graphs with exactly three non-essential edges. As a consequence, with the exception of the members of 10 classes of graphs, all 3-connected graphs have at least four non-essential edges.Acknowledgments The first author was partially supported by grants from the National Security Agency. The second author was partially supported by the Office of Naval Research under Grant No. N00014-01-1-0917.1991 Mathematics Subject Classification: 05C40Final version received: October 30, 2003     

Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465-473; H. Jiang, J. Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187-193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245-256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217-228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74-87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434-438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k?5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K₆. For a k-connected graph G such that end vertices of any edge of G have at most k-3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K₆ by a number of θ⁺-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K_k+1 or the graph H_k/2+1 obtained from K_k+2 by removing k/2+1 disjoint edges, and, if k is odd, G is isomorphic to K_k+1.     

An Extremal Problem on Contractible Edges in 3-Connected Graphs

An edge e in a 3-connected graph G is contractible if the contraction G/e is still 3-connected. The existence of contractible edges is a very useful induction tool. Let G be a simple 3-connected graph with at least five vertices. Wu [7] proved that G has at most vertices that are not incident to contractible edges. In this paper, we characterize all simple 3-connected graphs with exactly vertices that are not incident to contractible edges. We show that all such graphs can be constructed from either a single vertex or a 3-edge-connected graph (multiple edges are allowed, but loops are not allowed) by a simple graph operation. Research partially supported by an ONR grant under grant number N00014-01-1-0917     

A note on the signed edge domination number in graphs

Let G=(V(G),E(G)) be a graph. A function f:E(G)→{+1,−1} is called the signed edge domination function (SEDF) of G if ∑_e^′N[e]f(e^′)≥1 for every eE(G). The signed edge domination number of G is defined as is a SEDF of G}. Xu [Baogen Xu, Two classes of edge domination in graphs, Discrete Applied Mathematics 154 (2006) 1541–1546] researched on the edge domination in graphs and proved that for any graph G of order n(n≥4). In the article, he conjectured that: For any 2-connected graph G of order n(n≥2), . In this note, we present some counterexamples to the above conjecture and prove that there exists a family of k-connected graphs G_m,k with .     

Removable Edges and Chords of Longest Cycles in 3-Connected Graphs

We verify two special cases of Thomassen’s conjecture of 1976 stating that every longest cycle in a 3-connected graph contains a chord.We prove that Thomassen’s conjecture is true for two classes of 3-connected graphs that have a bounded number of removable edges on or off a longest cycle. Here an edge e of a 3-connected graph G is said to be removable if G – e is still 3-connected or a subdivision of a 3-connected (multi)graph.We give examples to showthat these classes are not covered by previous results.     

Removable edges in cyclically 4-edge-connected cubic graphs

We consider various ways of obtaining smaller cyclically 4-edge-connected cubic graphs from a given such graph. In particular, we consider removable edges: an edgee of a cyclically 4-edge-connected cubic graphG is said to be removable ifG is also cyclically 4-edge-connected, whereG is the cubic graph obtained fromG by deletinge and suppressing the two vertices of degree 2 created by the deletion. We prove that any cyclically 4-edge-connected cubic graphG with at least 12 vertices has at least 1/5(|E(G)| + 12) removable edges, and we characterize the graphs with exactly 1/5(|E(G)| + 12) removable edges.This work was carried out while the first author held a Niels Bohr Fellowship from the Royal Danish Academy of Sciences.     

The number of contractible edges in 3-connected graphs

An edge of a 3-connected graph is calledcontractible if its contraction results in a 3-connected graph. Ando, Enomoto and Saito proved that every 3-connected graph of order at least five has |G|/2 or more contractible edges. As another lower bound, we prove that every 3-connected graph, except for six graphs, has at least (2|E(G)| + 12)/7 contractible edges. We also determine the extremal graphs. Almost all of these extremal graphsG have more than |G|/2 contractible edges.     

Claw Conditions for Heavy Cycles in Weighted Graphs

A graph is called a weighted graph when each edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d^w(v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H. In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. A 2-connected weighted graph G contains either a Hamilton cycle or a cycle of weight at least c, if G satisfies the following conditions: In every induced claw or induced modified claw F of G, (1) max{d^w(x),d^w(y)} c/2 for each non-adjacent pair of vertices x and y in F, and (2) all edges of F have the same weight.     

Connectivity keeping edges in graphs with large minimum degree

The old well-known result of Chartrand, Kaugars and Lick says that every k-connected graph G with minimum degree at least 3k/2 has a vertex v such that G−v is still k-connected. In this paper, we consider a generalization of the above result [G. Chartrand, A. Kaigars, D.R. Lick, Critically n-connected graphs, Proc. Amer. Math. Soc. 32 (1972) 63–68]. We prove the following result:Suppose G is a k-connected graph with minimum degree at least 3k/2+2. Then G has an edge e such that G−V(e) is still k-connected.The bound on the minimum degree is essentially best possible.     

Reductions of 3-connected graphs with minimum degree at least four

We show that if G is a 3-connected graph of minimum degree at least 4 and with |V (G)| ≥ 7 then one of the following is true: (1) G has an edge e such that G/e is a 3-connected graph of minimum degree at least 4; (2) G has two edges uv and xy with ux, vy, vx ∈ E(G) such that the graph G/uv/xy obtained by contraction of edges uv and xy in G is a 3-connected graph of minimum degree at least 4; (3) G has a vertex x with N(x) = {x₁, x₂, x₃, x₄} and x₁x₂, x₃x₄ ∈ E(G) such that the graph (G ? x)/x₁x₂/x₃x₄ obtained by contraction of edges x₁x₂ and x₃x₄ in G – x is a 3-connected graph of minimum degree at least 4.

Each of the three reductions is necessary: there exists an infinite family of 3- connected graphs of minimum degree not less than 4 such that only one of the three reductions may be performed for the members of the family and not the two other reductions.     

A new lower bound on the number of trivially noncontractible edges in contraction critical 5-connected graphs

An edge of a k-connected graph is said to be k-contractible if its contraction results in a k-connected graph. A k-connected non-complete graph with no k-contractible edge, is called contraction critical k-connected. An edge of a k-connected graph is called trivially noncontractible if its two end vertices have a common neighbor of degree k. Ando [K. Ando, Trivially noncontractible edges in a contraction critically 5-connected graph, Discrete Math. 293 (2005) 61-72] proved that a contraction critical 5-connected graph on n vertices has at least n/2 trivially noncontractible edges. Li [Xiangjun Li, Some results about the contractible edge and the domination number of graphs, Guilin, Guangxi Normal University, 2006 (in Chinese)] improved the lower bound to n+1. In this paper, the bound is improved to the statement that any contraction critical 5-connected graph on n vertices has at least trivially noncontractible edges.     

Contractible edges inn-connected graphs with minimum degree greater than or equal to [5n/4]

It is proved that ifG is ann-connected graph with minimum degree greater than or equal to [5n/4],n 4, thenG has an edgee such that the graph obtained fromG by contractinge is stilln-connected.Dedicated to Professor Nagayoshi Iwahori on his 60th birthday     

Contractible edges in non-separating cycles

An edge of ak-connected graph is said to bek-contractible if the contraction of the edge results in ak-connected graph. We prove that every triangle-freek-connected graphG has an induced cycleC such that all edges ofC arek-contractible and such thatG–V(C) is (k–3)-connected (k4). This result unifies two theorems by Thomassen [5] and Egawa et. al. [3].Dedicated to Professor Toshiro Tsuzuku on his sixtieth birthday     

Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes

We define an analogue of Schnyder's tree decompositions for 3-connected planar graphs. Based on this structure we obtain: Let G be a 3-connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f–1)×(f–1) grid. Let G be a 3-connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3.The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.     

The 3-Connected Graphs with Exactly k Non-Essential Edges

Let G be a simple 3-connected graph. An edge e of G is essential if neither the deletion G\ e nor the contraction G/e is both simple and 3-connected. In this study, we show that all 3-connected graphs with k(k ≥ 2) non-essential edges can be obtained from a wheel by three kinds of operations which defined in the paper.     

A property of weighted graphs without induced cycles of nonpositive weights

It has been conjectured [B. Xu, On signed cycle domination in graphs, Discrete Math. 309 (4) (2009) 1007–1012] that if there is a mapping from the edge set of a 2-connected graph G to {−1,1} such that for each induced subgraph, that is a cycle, the sum of all numbers assigned to its edges by this mapping is positive, then the number of all those edges of G to which 1 is assigned, is more than the number of all other edges of G. This conjecture follows from the main result of this note: If a mapping assigns integers as weights to the edges of a 2-connected graphGsuch that for each edge, its weight is not more than 1 and for each cycle which is an induced subgraph ofG, the sum of all weights of its edges is positive, then the sum of all weights of the edges ofGalso is positive. A simple corollary of this result is the following: If?is a mapping from the edge set of a 2-connected graphGto a set of real numbers such that for each cycleCofG, ∑_e∈E(C)?(e)>0, then∑_e∈E(G)?(e)also is positive.