Edges In 3-connected Graphs Research Articles

Contractible Edges and Peripheral Cycles in 3-Connected Graphs

Peripheral cycles (induced non-separating cycles) in a general 3-connected graph are analogous to the faces of a polyhedron. Using the works of various authors, this paper explores the distribution of contractible edges in 3-connected graphs as needed to prove a major result originally by Tutte: each edge in a 3-connected graph is part of at least 2 peripheral cycles that share only the edge and its end vertices. A complete, alternative proof of this theorem is provided. The inductive step is generalized into a new independent lemma, which states that each edge in a 3-connected graph with a non-adjacent contractible edge has at least as many peripheral cycles as in the contracted one.

Contractible Edges and Removable Edges in 3-Connected Graphs

In this paper we prove that every spanning tree of a 3-connected 3-regular graph, except for $$K_4$$ and $$K_2\, \Box \,K_3$$, contains at least two contractible edges, and that every spanning tree of a minimally 3-connected graph, except for the wheel graphs, contains at least one contractible edge. Moreover, we show that every maximum matching of a 3-connected graph of order at least 6 that does not contain the maximal semiwheel graphs avoids at least four removable edges; every maximum matching of a 3-connected graph avoids at least two removable edges.

Contractible edges in 3-connected graphs that preserve a minor

Let G be a 3-connected graph with a 3-connected (or sufficiently small) simple minor H. We establish that G has a forest F with at least ⌈(|G|−|H|+1)/2⌉ edges such that G/e is 3-connected with an H-minor for each e∈E(F). Moreover, we may pick F with |G|−|H| edges provided G is triangle-free. These results are sharp. Our result generalizes a previous one by Ando et al., which establishes that a 3-connected graph G has at least ⌈|G|/2⌉ contractible edges. As another consequence, each triangle-free 3-connected graph has an spanning tree of contractible edges. Our results follow from a more general theorem on graph minors, a splitter theorem, which is also established here.

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 K 6 . 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 K 6 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 [InlineMediaObject not available: see fulltext.] vertices that are not incident to contractible edges. In this paper, we characterize all simple 3-connected graphs with exactly [InlineMediaObject not available: see fulltext.] 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.

Contractible Elements in Graphs and Matroids

Let G be a simple 3-connected graph with at least five vertices. Tutte [13] showed that G has at least one contractible edge. Thomassen [11] gave a simple proof of this fact and showed that contractible edges have many applications. In this paper, we show that there are at most $\frac{|V(G)|}{5}$ vertices that are not incident to contractible edges in a 3-connected graph G. This bound is best-possible. We also show that if a vertex v is not incident to any contractible edge in G, then v has at least four neighbours having degree three, and each such neighbour is incident to exactly two contractible edges. We give short proofs of several results on contractible edges in 3-connected graphs as well. We also study the contractible elements for k-connected matroids. We partially solve an open problem for regular matroids.

On Non-Essential Edges in 3-Connected Graphs

An edge e in a simple 3-connected graph is deletable (simple-contractible) if the deletion G\e (contraction G/e) is both simple and 3-connected. Suppose a, b, and c are three non-negative integers. If there exists a simple 3-connected graph with exactly a edges which are deletable but not simple-contractible, exactly b edges which are simple-contractible but not deletable, and exactly c edges which are both deletable and simple-contractible, then we call the triple (a, b, c) realizable, and such a graph is said to be an (a, b, c)-graph. Tutte's Wheels Theorem says the only (0, 0, 0)-graphs are the wheels. In this paper, we characterize the (a, b, c) realizable triples for which at least one of a + b≤2, c=0, and c≥16 holds.

The Number of Removable Edges in 3-Connected Graphs

An edge of a 3-connected graph G is said to be removable if G − e is a subdivision of a 3-connected graph. Holton et al. (1990) proved that every 3-connected graph of order at least five has at least ⌈(| G |+10)/6⌉ removable edges. In this paper, we prove that every 3-connected graph of order at least five, except the wheels W 5 and W 6 , has at least (3 | G |+18)/7 removable edges. We also characterize the graphs with (3 |G|+18)/7 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 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.

Contractible edges in 3-connected graphs

By Tutte's constructive characterization of 3-connected graphs ( Indag. Math. 23 (1961) , 441–455), we see that every 3-connected graph of order at least five has an edge whose contraction results in a 3-connected graph. We call such an edge a contractible edge and study the distribution of contractible edges in 3-connected graphs. As a consequence, we prove that every 3-connected graph of order at least five has [ |G| 2 ] or more contractible edges and determine the graphs which attain the equality.