3-connected Planar Research Articles

On unique determination of conformal type for domains in Euclidean spaces

This survey is devoted to discussion of problems of the unique determination of conformal type for domains in Euclidean spaces. It consists of three parts. In the first of them, we consider results on the problem of the unique determination for (generally speaking) nonconvex domains in R n , where n ≥ 4. The second part is devoted to study of problems on unique determination of convex polyhedral domains in three-dimensional Euclidean space by relative conformal moduli of boundary condensers. Finally, in the third part, we study problems of unique determination of 3-connected plane domains by relative conformal moduli of pairs of boundary components.

Connectivity and $$W_v$$ W v -Paths in Polyhedral Maps on Surfaces

The $W_v$-Path Conjecture due to Klee and Wolfe states that any two vertices of a simple polytope can be joined by a path that does not revisit any facet. This is equivalent to the well-known Hirsch Conjecture. Klee proved that the $W_v$-Path Conjecture is true for all 3-polytopes (3-connected plane graphs), and conjectured even more, namely that the $W_v$-Path Conjecture is true for all general cell complexes. This general $W_v$-Path Conjecture was verified for polyhedral maps on the projective plane and the torus by Barnette, and on the Klein bottle by Pulapaka and Vince. Let $G$ be a graph polyhedrally embedded in a surface $\Sigma$, and $x, y$ be two vertices of $G$. In this paper, we show that if there are three internally disjoint $(x,y)$-paths which are homotopic to each other, then there exists a $W_v$-path joining $x$ and $y$. For every surface $\Sigma$, define a function $f(\Sigma)$ such that if for every graph polyhedrally embedded in $\Sigma$ and for a pair of vertices $x$ and $y$ in $V(G)$, the local connectivity $\kappa_G(x,y) \ge f(\Sigma)$, then there exists a $W_v$-path joining $x$ and $y$. We show that $f(\Sigma)=3$ if $\Sigma$ is the sphere, and for all other surfaces $3-\tau(\Sigma)\le f(\Sigma)\le 9-4\chi(\Sigma)$, where $\chi(\Sigma)$ is the Euler characteristic of $\Sigma$, and $\tau(\Sigma)=\chi(\Sigma)$ if $\chi(\Sigma)< -1$ and 0 otherwise. Further, if $x$ and $y$ are not cofacial, we prove that $G$ has at least $\kappa_G(x,y)+4\chi(\Sigma)-8$ internally disjoint $W_v$-paths joining $x$ and $y$. This bound is sharp for the sphere. Our results indicate that the $W_v$-path problem is related to both the local connectivity $\kappa_G(x,y)$, and the number of different homotopy classes of internally disjoint $(x,y)$-paths as well as the number of internally disjoint $(x,y)$-paths in each homotopy class.

Competition Numbers of a Kind of Pseudo-Halin Graphs

For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and chara-cterizing a graph by its competition number has been one of important research problems in the study of competition graphs. A 2-connected planar graph G with minimum degree at least 3 is a pseudo-Halin graph if deleting the edges on the boundary of a single face f0 yields a tree. It is a Halin graph if the vertices of f0 all have degree 3 in G. In this paper, we compute the competition numbers of a kind of pseudo-Halin graphs.

Long induced paths in graphs

We prove that every 3-connected planar graph on n vertices contains an induced path on Ω(logn) vertices, which is best possible and improves the best known lower bound by a multiplicative factor of loglogn. We deduce that any planar graph (or more generally, any graph embeddable on a fixed surface) with a path on n vertices, also contains an induced path on Ω(logn) vertices. We conjecture that for any k, there is a positive constant c(k) such that any k-degenerate graph with a path on n vertices also contains an induced path on Ω((logn)c(k)) vertices. We provide examples showing that this order of magnitude would be best possible (already for chordal graphs), and prove the conjecture in the case of interval graphs.

Refined weight of edges in normal plane maps

The weight w(e) of an edge e in a normal plane map (NPM) is the degree-sum of its end-vertices. An edge e=uv is of type (i,j) if d(u)≤i and d(v)≤j. In 1940, Lebesgue proved that every NPM has an edge of one of the types (3,11), (4,7), or (5,6), where 7 and 6 are best possible. In 1955, Kotzig proved that every 3-connected planar graph has an edge e with w(e)≤13, which bound is sharp. Borodin (1989), answering Erdős’ question, proved that every NPM has either a (3,10)-edge, or (4,7)-edge, or (5,6)-edge.A vertex is simplicial if it is completely surrounded by 3-faces. In 2010, Ferencová and Madaras conjectured (in different terms) that every 3-polytope without simplicial 3-vertices has an edge e with w(e)≤12. Recently, we confirmed this conjecture by proving that every NPM has either a simplicial 3-vertex adjacent to a vertex of degree at most 10, or an edge of types (3,9), (4,7), or (5,6).By a k(ℓ)-vertex we mean a k-vertex incident with precisely ℓ triangular faces. The purpose of our paper is to prove that every NPM has an edge of one of the following types: (3(3),10), (3(2),9), (3(1),7), (4(4),7), (4(3),6), (5(5),6), or (5,5), where all bounds are best possible. In particular, this implies that the bounds in (3,10), (4,7), and (5,6) can be attained only at NPMs having a simplicial 3-, 4-, or 5-vertex, respectively.

Erratum to: “On unique determination of 3-connected plane domains by relative conformal moduli of pairs of boundary components”

The boundary values of conformal mappings of plane finitely connected domains are studied. A complete description of the boundary values of such mappings in terms of the moduli (extremal lengths) of pairs of boundary components of domains is obtained in the case where the number of boundary components is not larger than three.

Lower and upper bounds for long induced paths in 3-connected planar graphs

Let G be a 3-connected planar graph with n vertices and let p(G) be the maximum number of vertices of an induced subgraph of G that is a path. Substantially improving previous results, we prove that p(G)≥log⁡n12log⁡log⁡n. To demonstrate the tightness of this bound, we notice that the above inequality implies p(G)∈Ω((log2⁡n)1−ε), where ε is any positive constant smaller than 1, and describe an infinite family of planar graphs for which p(G)∈O(log⁡n). As a byproduct of our research, we prove a result of independent interest: Every 3-connected planar graph with n vertices contains an induced subgraph that is outerplanar and connected and that contains at least n3 vertices. The proofs in the paper are constructive and give rise to O(n)-time algorithms.

On unique determination of 3-connected plane domains by relative conformal moduli of pairs of boundary components

On unique determination of 3-connected plane domains by relative conformal moduli of pairs of boundary components

Gray coding cubic planar maps

The idea of (combinatorial) Gray codes is to list objects in question in such a way that two successive objects differ in some pre-specified small way. In this paper, we utilize β-description trees to cyclicly Gray code three classes of cubic planar maps, namely, bicubic planar maps, 3-connected cubic planar maps, and cubic non-separable planar maps.

Generating functions for stoichiometry and structure of single- and double-layer sheet-silicates

AbstractTwo-dimensional nets may be used to generate the stoichiometry and structure of single-layer and double-layer sheet-silicate minerals. Many sheet-silicate minerals are based on the 3-connected plane nets 63, 4.82, (4.6.8)2(6.82)1and (52.8)1(5.82)1, and some more complicated nets, e.g. (5.6.7)4(5.72)1(62.7)1, (4.122)2(42.12)1, (52.8)1(5.62)1(5.6.8)2(62.8)1,have one or two representative structures. Many complicated sheet-silicate minerals are based on sheets of 2-, 3- and 4-connected tetrahedra that may be developed from 3- and 4-connected plane nets by a series of oikodoméic operations on 3- or 4-connected nets that change the topologyof the parent net. There are three classes of oikodoméic operations: (1) insertion of 2- and 3-connected vertices into 3- and 4-connected plane nets; (2,3) replication of single-layer sheets by topological mirror or two-fold-rotation operators, and condensation of the resulting twosingle-layer sheets to form double-layer sheets. The topological aspects of these sheet structures may be described by functions that express stoichiometry in terms of tetrahedron connectivities (formula-generating functions) and functions that associate these formula-generating functionswith specific two-dimensional nets. Using these functions, we may generate formulae and structural arrangements of single-layer and double-layer silicate structures with specific local and long-range topological features.

On k-greedy routing algorithms

Let G=(V,E) and G′=(V′,E′) be two graphs. A k-inverse-adjacency-preserving mapping ρ from G to G′ is a one-to-many and onto mapping from V to V′ satisfying the following: (1) Each vertex v∈V in G is mapped to a non-empty subset ρ(v)⊂V′ in G′, the cardinality of ρ(v) is at most k; (2) if u≠v, then ρ(u)∩ρ(v)=∅; and (3) for any u′∈ρ(u) and v′∈ρ(v), if (u′,v′)∈E′, then (u,v)∈E. A vertex u′ in ρ(u) is called a virtual location for u.Let ρ be a k-inverse-adjacency-preserving-mapping (k-IAPM for short) from G to G′. Let δ be a greedy drawing of G′ into a metric space M. Consider a message from u to be delivered to v in G. Using the k-IAPM ρ from G to G′, intuitively, one can treat the message to be routed as if it were from one virtual location u′ for u to one virtual location v′ for v, except that all the virtual locations of a vertex u were identified with each other (which can be thought as instantaneously synchronized mirror sites for a particular website, for example). Then a routing path P′ can be computed from u′ to v′ while identifying all virtual locations for any vertex in G. Since ρ inversely preserves adjacency from G′ to G, such a routing path P′ corresponds to exactly one routing path P in G, which connects u to v.In this paper, we formalize the above intuition into a concept which we call k-greedy routing algorithm for a graph G with n vertices, where k refers to the maximum number of virtual locations any vertex of G can have. Using this concept, the result presented in [18] can be rephrased as a 3-greedy routing algorithm for 3-connected plane graphs, where the virtual coordinates used are from 1 to 2n−2. In this paper, we present a 2-greedy routing algorithm for 3-connected plane graphs, where each vertex uses at least one but at most two virtual locations numbered from 1 to 2n−1. For the special case of plane triangulations (in a plane triangulation, every face is a triangle, including the exterior face), the numbers used are further reduced to from 1 to ⌊5n+13⌋. Hence, there are at least ⌈n−13⌉ vertices that use only one virtual location.

Decomposing plane cubic graphs

It was conjectured by Hoffmann-Ostenhof that the edge set of every cubic graph can be decomposed into a spanning tree, a matching and a family of cycles. We prove the conjecture for 3-connected cubic plane graphs and 3-connected cubic graphs on the projective plane. Our proof provides a polynomial time algorithm to find the decomposition for 3-connected cubic plane graphs.

Verification on a Given Point Set for a Cubic Plane Graph

Cubic graph, where all vertices have degree three can be associated as 3-regular graph and trivalent graph. Let a point P be the given point set, with a stipulation of n ≥ 4 points in the regular plane such that n is even in general situation. It has been probed that the problem whether there is a straight-line cubic plane graph on P. Algorithm is not known for polynomialtime of a problem. With research on the reduction to the existence of the convex hull of P along with the certain diagonals of the boundary cycle, the polynomial-time algorithm is proposed that checks for 2-connected cubic plane graphs. The algorithm is constructive and runs in time complexity of O (n3). It is also shown that which graph structure can be expected when there is a cubic plane graph on P; e. g., a cubic plane graph on P implies a connected cubic plane graph on P, and a 2-connected cubic plane graph on P implies a 2-connected cubic plane graph on P that contains the boundary cycle of P.

Sufficient Condition and Algorithm for Hamiltonian in 3-Connected 3-Regular Planar Bipartite Graph

graph G (V, E) is said to be Hamiltonian if it contains a spanning cycle. The spanning cycle is called a Hamiltonian cycle of G and G is said to be a Hamiltonian graph. A Hamiltonian path is a path that contains all the vertices in V (G) but does not return to the vertex in which it began. In this paper, we study Hamiltonicity of 3-connected, 3-regular planar bipartite graph G with partite sets V=M  N. We shall prove that G has a Hamiltonian cycle if G is balanced with M = N. For that we present an algorithm for a bipartite graph KM,N where M>3, N>3 and M,N both are even to possess a Hamiltonian cycle. In particular, we also prove a theorem for S proper subset (M or N) of V the number of components W (G-S) = S implies the graph has a Hamiltonian path.

Wheel-free planar graphs

A wheel is a graph formed by a chordless cycle C and a vertex u not in C that has at least three neighbors in C. We prove that every 3-connected planar graph that does not contain a wheel as an induced subgraph is either a line graph or has a clique cutset. We prove that every planar graph that does not contain a wheel as an induced subgraph is 3-colorable.

A mathematical programming model for computing the Fries number of a fullerene

A fullerene graph is a cubic 3-connected plane graph with pentagonal and hexagonal faces. The Fries number of a fullerene is the maximum number of benzene-like faces over all possible perfect matchings. The Fries number and its associated Kekulé structure of a fullerene play a key role in molecular energy and stability. In this paper we propose a binary integer linear programming and a quadratic programming model for determining the Fries number of a fullerene. Moreover, interior point approach, as one of the most robust optimization techniques, is implemented to find the optimal solution of the proposed quadratic programming problem in moderate computing time.

On Barnette’s conjecture and the $$H^{+-}$$ H + - property

Let $$\mathcal{B}$$B denote the class of all 3-connected cubic bipartite plane graphs. A conjecture of Barnette states that every graph in $$\mathcal{B}$$B has a Hamilton cycle. A cyclic sequence of big faces is a cyclic sequence of different faces, each bounded by at least six edges, such that two faces from the sequence are adjacent if and only if they are consecutive in the sequence. Suppose that $${F_1, F_2, F_3}$$F1,F2,F3 is a proper 3-coloring of the faces of $$G^*\in \mathcal{B}$$G??B. We prove that if every cyclic sequence of big faces of $$G^*$$G? has a face belonging to $$F_1$$F1 and a face belonging to $$F_2$$F2, then $$G^*$$G? has the following properties: $$H^{+-}$$H+-: If any two edges are chosen on the same face, then there is a Hamilton cycle through one and avoiding the other, $$H^{--}$$H--: If any two edges are chosen which are an even distance apart on the same face, then there is a Hamilton cycle that avoids both. Moreover, let $$X$$X and $$Y$$Y partition the set of big faces of $$G^*$$G? such that all such faces of $$F_1$$F1 are in $$X$$X and all such faces of $$F_2$$F2 are in $$Y$$Y. We prove that if every cyclic sequence of big faces has a face belonging to $$X$$X and a face belonging to $$Y$$Y, then $$G^*$$G? has a Hamilton cycle.

Connectivity and formula-generating functions for sheet-silicate minerals

Silicate sheets may be described by two-dimensional nets in which the vertices of the net are occupied by tetrahedra, and the edges of the net represent linkages between tetrahedra. A plane net must contain 3-connected vertices, but not all vertices need to be 3-connected. Simple silicate structures may thus be generated from simple 3-connected plane nets (e.g. 63, 4.82, 4.6.8, (4.6.8)2(6.82)1, etc.). More complicated silicate nets may be generated by various "building operations": (1) Insertion: insertion of 2- and 4-connected vertices into 3-connected plane nets; (2) Repetition: generation of double (or triple) nets by topological symmetry operations that retain transitivity at the junction between the repeated elements. Diversity is also introduced within the sheets of tetrahedra by [1] adjacent apical tetrahedron vertices pointing in the same or different directions, and [2] by folding of the sheets. For simple structures, net type strongly affects the stoichiometry of the resultant structure as the unit cells of the various nets are of different sizes (and shapes), although the stoichiometry may also be affected by non-tetrahedral components. Building operations strongly affect the stoichiometry of the resultant sheet, and this effect may be quantified. We define a formula-generating function F(k,l,...) that generates the formula of a sheet with specific topological features denoted by the indices k,l,... . A simple 3-connected net results in sheets of the form (T2O5)n where n denotes the number of (T2O5)n in the unit cell of the underlying net (for 63, n = 1; for 4.82, n = 2; for (4.6.8)2(6.82)1, n = 3, etc). Plane nets with k 3-connected vertices and l inserted 2-connected vertices result in sheets of the form [T(k+l) O(2.5k+3l)], where (...) are subscripted. Single- and double-sheet structures may be generated from the function F(k,l) = T(N{k+l}) O(N{3k+2.5l}-n{N-1}) where N = 1 and 2 for single- and double-sheets, respectively, and (...) are subscripted.

Spanning closed walks and TSP in 3-connected planar graphs

In this paper, we prove the following theorem, which is motivated by two different contexts independently, namely graph theory and combinatorial optimization. Our proof is constructive (and purely combinatorial) in the sense that there is an O(n2)-time algorithm to construct such a walk in a given 3-connected planar graph. We shall construct an example that shows that the upper 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.

Family of[[6k,2k,2]]codes for practical and scalable adiabatic quantum computation

In this work, we introduce a family of $[[6k,2k,2]]$ codes designed specifically to be compatible with adiabatic quantum computation. These codes support computationally universal sets of weight 2 logical operators and are particularly well suited for implementing dynamical decoupling error suppression. For Hamiltonians embeddable on a planar graph of fixed degree, our encoding maintains a planar connectivity graph and increases the graph degree by only two.