Parity of Cycles Containing Specified Edges

We consider the following problem which is motivated by two different contexts independently, namely graph theory and combinatorial optimization.Given a 3-connected planar graph G with n vertices, is there a spanning closed walk W with at most 4n/3 edges?In graph theory, the above question is motivated by the famous hamiltonian result by Tutte in 1956 which says that every 4-connected planar graph is hamiltonian (a simpler proof is given by Thomassen in 1983). What happens if we relax the 4-connectivity? There is a 3-connected planar graph that is not hamiltonian, but how about a spanning close walk (which is exactly a traveling salesman tour. Sometimes such a walk is called hamiltonian walk)? How many edges are necessary to cover all the vertices of a 3-connected planar graph by a closed walk? This is exactly the above question.In combinatorial optimization, the famous traveling salesman problem in metric graphs is one of most fundamental NP-hard optimization problems. In spite of a vast amount of research several important questions remain open. In particular, the best known upper bound is not believed to be best possible. A promising direction to improve this approximation guarantee, has long been to understand the power of a linear program known as the Held-Karp relaxation [11]. On the one hand, the best lower bound on its integrality gap (for the symmetric case) is 4/3 and indeed the famous (so called 4/3-)conjecture said that this lower bound would be tight [10]. Goemans pointed out that there is a planar graph that achieves this bound. So he brought attention to the above question, i.e, the famous 4/3-conjecture is always true for 3-connected planar graphs.We prove the above problem in the following strong form;Given a circuit graph (which is obtained from a 3-connected planar graph by deleting one vertex) with n vertices, there is a spanning closed walk with at most 4(n − 1)/3 edges such that each edge is used at most twice.Moreover, our proof is constructive (and purely combinatorial) in a sense that there is an O(n2) algorithm to construct, given a 3-connected planar graph, such a walk.We shall construct an example that shows that the bound 4(n − 1)/3 is essentially tight. We also point out that 2-connected planar graphs may not have such a walk, as K2,n−2 shows.

For two integers m,n with m ≤ n, an [m, n]-factor F in a graph G is a spanning subgraph of G with m ≤ dF(v) ≤ n for all v ∈ V(F). In 1996, H. Enomoto et al. proved that every 3-connected planar graph G with dG(v) ≥ 4 for all v ∈ V(G) contains a [2,3]-factor. In this paper we extend their result to all 3-connected locally finite infinite planar graphs containing no unbounded faces.

Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular planar graph $G$ embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings of $G$ on each surface and some subgraphs of the dual of $G$ embedded on the sphere. These results enable us to give explicit bounds for the number of inequivalent embeddings of $G$ on each surface, and propose effective algorithms for enumerating and counting these embeddings.Comment: 19 pages, 12 figures

The Clar number of a (hydro)carbon molecule, introduced by Clar (The aromatic sextet, 1972), is the maximum number of mutually disjoint resonant hexagons in the molecule. Calculating the Clar number can be formulated as an optimization problem on 2-connected plane graphs. Namely, it is the maximum number of mutually disjoint even faces a perfect matching can simultaneously alternate on. It was proved by Abeledo and Atkinson (Linear Algebra Appl 420(2):441–448, 2007) that the Clar number can be computed in polynomial time if the plane graph has even faces only. We prove that calculating the Clar number in general 2-connected plane graphs is \(\mathsf {NP}\)-hard. We also prove \(\mathsf {NP}\)-hardness of the maximum independent set problem for 2-connected plane graphs with odd faces only, which may be of independent interest. Finally, we give an exact algorithm that determines the Clar number of a given 2-connected plane graph. The algorithm has a polynomial running time if the length of the shortest odd join in the planar dual graph is fixed, which gives an efficient algorithm for some fullerene classes, such as carbon nanotubes.

Geometric routing by using virtual locations is an elegant way for solving network routing problems. Greedy routing, where a message is simply forwarded to a neighbor that is closer to the destination, is a simple form of geometric routing. Papadimitriou and Ratajczak conjectured that every 3-connected plane graph has a greedy drawing in the $\mathcal R^2$ plane [10]. Leighton and Moitra settled this conjecture positively in [9]. However, their drawings have two major drawbacks: (1) their drawings are not necessarily planar; and (2) Ω(n logn ) bits are needed to represent the coordinates of their drawings, which is too large for routing algorithms for wireless networks. Recently, He and Zhang [8] showed that every triangulated plane graph has a succinct (using O (logn ) bit coordinates) greedy drawing in $\mathcal R^2$ plane with respect to a metric function derived from Schnyder realizer. However, their method fails for 3-connected plane graphs. In this paper, we show that every 3-connected plane graph has drawing in the $\mathcal R^2$ plane, that is succinct, planar, strictly convex, and is greedy with respect to a metric function based on parameters derived from Schnyder wood.

The cyclic vertex (resp., edge) connectivity of a graph [Formula: see text], denoted by [Formula: see text] (resp., [Formula: see text]), is the minimum number of vertices (resp., edges) whose removal from [Formula: see text] results in a disconnected graph and at least two remaining components contain cycles. Thus, to determine the exact values of [Formula: see text] and [Formula: see text] is important in the reliability assessment of interconnection networks. However, the study of the cyclic vertex (edge) connectivity is less involved. In this paper, we determine the cyclic vertex (edge) connectivity of the burnt pancake graphs [Formula: see text] which is the Cayley graph of the group of signed permutations using prefix reversals as generators. By exploring the combinatorial properties and fault-tolerance of [Formula: see text], we show [Formula: see text] and [Formula: see text] for [Formula: see text]. Moreover, we determine that [Formula: see text] for [Formula: see text].

Prescribed edges and forbidden edges for a cycle in a planar graph

Spanning closed walks and TSP in 3-connected planar graphs

The decomposition of a connected graph by the set of its cut-vertices, sometimes called the \\"block decomposition\\" or \\"block tree\\" of a graph, is a well known and basic concept in graph theory. This decomposition, however, does not provide meaningful information when applied to a k-connected graph for k > 1. There has been a number of attempts to generalize the construction of the block decomposition of a graph for the case of k-connected graphs. Notably, Tutte constructed a tree that describes the mutual arrangement of 2-cutsets in a 2-connected graph. This decomposition has some similarities to the block decomposition of a connected graph. In other works, a block of a k-connected graph was defined as a maximal (k+1)-connected subgraph. Karpov described the decomposition of a k-connected graph by the set of k-cutsets that are not separated by any other k-cutset of the graph. Karpov also described some special properties of his decomposition for the case of a 2-connected graph. The decompositions defined by Karpov and Tutte for the case of a 2-connected graph share some similarities. In this work, we present a self-contained description of Karpov\\'s decomposition. We also present some applications to the study of planarity, the chromatic number, critically 2-connected graphs, and the partition of certain 2-connected graphs into three connected subgraphs.

Dynamic Complexity (as introduced by Patnaik and Immerman [14]) tries to express how hard it is to update the solution to a problem when the input is changed slightly. It considers the changes required to some stored data structure (possibly a massive database) as small quantities of data (or a tuple) are inserted or deleted from the database (or a structure over some vocabulary). The main difference from previous notions of dynamic complexity is that instead of treating the update quantitatively by finding the the time/space trade-offs, it tries to consider the update qualitatively, by finding the complexity class in which the update can be expressed (or made). In this setting, DynFO, or Dynamic First-Order, is one of the smallest and the most natural complexity class (since SQL queries can be expressed in First-Order Logic), and contains those problems whose solutions (or the stored data structure from which the solution can be found) can be updated in First-Order Logic when the data structure undergoes small changes.Etessami [7] considered the problem of isomorphism in the dynamic setting, and showed that Tree Isomorphism can be decided in DynFO. In this work, we show that isomorphism of Planar 3-connected graphs can be decided in DynFO + (which is DynFO with some polynomial precomputation). We maintain a canonical description of 3-connected Planar graphs by maintaining a database which is accessed and modified by First-Order queries when edges are added to or deleted from the graph. We specifically exploit the ideas of Breadth-First Search and Canonical Breadth-First Search to prove the results. We also introduce a novel method for canonizing a 3-connected planar graph in First-Order Logic from Canonical Breadth-First Search Trees.KeywordsShort PathPlanar GraphUndirected GraphDynamic ComplexityComplexity ClassThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On contractible and vertically contractible elements in 3-connected matroids and graphs

Given a weighting of all elements of a 2-connected plane graph G = (V,E, F ), let f(α) denote the sum of the weights of the edges and vertices incident with the face α and also the weight of α. Such an entire weighting is a proper face colouring provided that f(α) = f(β) for every two faces α and β sharing an edge. We show that for every 2-connected plane graph there is a proper face-colouring entire weighting with weights 1 through 4. For some families we improved 4 to 3.

Long cycles through specified vertices in a graph

The vertex connectivity of an m-edge n-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph, or leaves only a singleton vertex. In this paper, we give a reduction from the vertex connectivity problem to a set of maxflow instances. Using this reduction, we can solve vertex connectivity in (mα) time for any α ≥ 1, if there is a mα-time maxflow algorithm. Using the current best maxflow algorithm that runs in m4/3+o(1) time (Kathuria, Liu and Sidford, FOCS 2020), this yields a m4/3+o(1)-time vertex connectivity algorithm. This is the first improvement in the running time of the vertex connectivity problem in over 20 years, the previous best being an O(mn)-time algorithm due to Henzinger, Rao, and Gabow (FOCS 1996). Indeed, no algorithm with an o(mn) running time was known before our work, even if we assume an (m)-time maxflow algorithm. Our new technique is robust enough to also improve the best O(mn)-time bound for directed vertex connectivity to mn1−1/12+o(1) time

Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ Hamiltonian cycles. We complement this by proving that the same conclusion holds for planar 4-regular 3-connected graphs, although it does not hold for planar 4-regular 4-connected graphs by a result of Brinkmann and Van Cleemput [European J. Combin., 97 (2021), 103395], and that it holds for 4-regular graphs of connectivity 2 with the constant $144 < c$, which we believe to be minimal among all Hamiltonian 4-regular graphs of sufficiently large order. We then disprove a conjecture of Haythorpe by showing that for every nonnegative integer $k$ there is a 5-regular graph on $26 + 6k$ vertices with $2^{k+10} \cdot 3^{k+3}$ Hamiltonian cycles. We prove that for every $d \ge 3$ there is an infinite family of Hamiltonian 3-connected graphs with minimum degree $d$, with a bounded number of Hamiltonian cycles. It is shown that if a 3-regular graph $G$ has a unique longest cycle $C$, at least two components of $G - E(C)$ have an odd number of vertices on $C$, and that there exist 3-regular graphs with exactly two such components.