3-connected Planar Graphs Research Articles

Face irregular evaluations of family of grids

Let G = ( V , E , F ) be a connected plane graph. Let ρ : α ( V ) ∪ β ( E ) ∪ γ ( F ) → { 1 , 2 , … , k } be a k-labeling where α , β , γ ∈ { 0 , 1 } and ( α , β , γ ) ≠ ( 0 , 0 , 0 ) . The weight of a face f under ρ is defined as Wt ( f ) = α ∑ v ∼ f ρ ( v ) + β ∑ e ∼ f ρ ( e ) + γ ρ ( f ) . Then ρ is called a face irregular labeling of type ( α , β , γ ) if Wt ( f ) ≠ Wt ( g ) for any two faces f ≠ g . The face irregular strength of G is the smallest integer k such that G admits a face irreg ular k-labeling. In this paper we determine the face irregular strength of the grid graph P n + 1 □ P m + 1 under a labeling of type ( α , β , γ ) .

Cutting Barnette graphs perfectly is hard

A perfect matching cut is a perfect matching that is also a cutset, or equivalently, a perfect matching containing an even number of edges on every cycle. The corresponding algorithmic problem, Perfect Matching Cut, is known to be NP-complete in subcubic bipartite graphs [Le & Telle, TCS '22], but its complexity was open in planar graphs and cubic graphs. We settle both questions simultaneously by showing that Perfect Matching Cut is NP-complete in 3-connected cubic bipartite planar graphs or Barnette graphs. Prior to our work, among problems whose input is solely an undirected graph, only Distance-2 4-Coloring was known to be NP-complete in Barnette graphs. Notably, Hamiltonian Cycle would only join this private club if Barnette's conjecture were refuted.

Factors of HOMFLY polynomials

We study factorizations of HOMFLY polynomials of certain prime knots and oriented links. We begin with a computer analysis of prime knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for HOMFLY polynomials of oriented links associated to 2-connected plane graphs.

ON THE LOCATING CHROMATIC NUMBER OF DISJOINT UNION OF BUCKMINSTERFULLERENE GRAPHS

Let be a connected non-trivial graph. Let c be a proper vertex-coloring using k colors, namely . Let be a partition of induced by , where is the color class that receives the color . The color code, denoted by , is defined as , where for , and is the distance between two vertices and in G. If all vertices in have different color codes, then is called as the locating-chromatic -coloring of . The locating-chromatic number of , denoted by , is the minimum such that has a locating coloring. Let be the Buckminsterfullerene graph on vertices. Buckminsterfullerene graph is a 3-connected planar graph and a member of the fullerene graphs, representing fullerene molecules in chemistry. In this paper, we determine the locating chromatic number of the disjoint union of Buckminsterfullerene graphs, denoted by .

Using the graph p-distance coloring algorithm for partitioning atoms of some fullerenes

A molecular graph p-distance coloring is a partitioning of atoms in such a way that there are no path of distance less than or equal to p between the atoms of each set. A fullerene graph is a planar 3-connected cubic graph whose faces are pentagons and hexagons. In this paper we apply some existing coloring algorithms for partitioning atoms of fullerenes based on their distances from each other. We also use these algorithm for coloring faces of some fullerenes.

The polyomino graphs whose resonance graphs have a 1-degree vertex

A polyomino graph is a 2-connected plane graph such that each interior face is a square of side length one. The resonance graph of a polyomino graph P is a graph in which the vertices are the 1-factors of P and two vertices are adjacent provided their corresponding two 1-factors differ in exactly one unite square. In this paper, we determine the polyomino graphs whose resonance graph has a 1-degree vertex by constructing a 4-coordinate system. As consequences we determine the polyomino graphs with a forcing edge and an anti-forcing edge respectively.

Cyclic edge and cyclic vertex connectivity of [formula omitted]-fullerene graphs

Cyclic vertex connectivity cκ and cyclic edge connectivity cλ are two important kinds of conditional connectivity, which reflect the number of vertices or edges that can be removed before the graph is disconnected and at least two components contain a cycle, respectively. They have important applications in various networks such as computer networks or biochemical networks. In addition, a fullerene is a special kind of molecule in chemistry. A classic fullerene graph is a 3-connected cubic planar graph with only pentagonal and hexagonal faces. (4, 5, 6)-fullerene graphs are atypical fullerene graphs which also contain 4-faces. In this paper, we prove that cκ=cλ for (4,5,6)-fullerene graphs except for four exceptional graphs with order less than 16. We also give O(ν)-algorithms to determine the cyclic vertex connectivity and the cyclic edge connectivity of (4,5,6)-fullerene graphs.

Independence number and spectral radius of cactus graphs

Four Color Theorem implies that the independence number of every planar graph is at least n/4. A cactus graph is a connected planar graph whose block is either an edge or a cycle. In this paper, the lower bound for independence number of cactus graphs is improved to ⌈n/3⌉. Moreover, the maximum spectral radius of cactus graphs with fixed independence number is determined.

The Realizability of Theta Graphs as Reconfiguration Graphs of Minimum Independent Dominating Sets

Abstract The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted ℐ (G), is the graph whose vertices correspond to the i(G)-sets, and where two i(G)-sets are adjacent if and only if they differ by two adjacent vertices. Not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a source graph G such that H = ℐ (G). We consider a class of graphs called “theta graphs”: a theta graph is the union of three internally disjoint nontrivial paths with the same two distinct end vertices. We characterize theta graphs that are i-graph realizable, showing that there are only finitely many that are not. We also characterize those line graphs and claw-free graphs that are i-graphs, and show that all 3-connected cubic bipartite planar graphs are i-graphs.

The Structural Properties of (2, 6)-Fullerenes

A (2,6)-fullerene F is a 2-connected cubic planar graph whose faces are only 2-length and 6-length. Furthermore, it consists of exactly three 2-length faces by Euler’s formula. The (2,6)-fullerene comes from Došlić’s (k,6)-fullerene, a 2-connected 3-regular plane graph with only k-length faces and hexagons. Došlić showed that the (k,6)-fullerenes only exist for k=2, 3, 4, or 5, and some of the structural properties of (k,6)-fullerene for k=3, 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of (2,6)-fullerene. We discover that the edge-connectivity of (2,6)-fullerenes is 2. Every (2,6)-fullerene is 1-extendable, but not 2-extendable (F is called n-extendable (|V(F)|≥2n+2) if any matching of n edges is contained in a perfect matching of F). F is said to be k-resonant (k≥1) if the deleting of any i (0≤i≤k) disjoint even faces of F results in a graph with at least one perfect matching. We have that every (2,6)-fullerene is 1-resonant. An edge set, S, of F is called an anti−Kekulé set if F−S is connected and has no perfect matchings, where F−S denotes the subgraph obtained by deleting all edges in S from F. The anti−Kekulé number of F, denoted by ak(F), is the cardinality of a smallest anti−Kekulé set of F. We have that every (2,6)-fullerene F with |V(F)|>6 has anti−Kekulé number 4. Further we mainly prove that there exists a (2,6)-fullerene F having fF hexagonal faces, where fF is related to the two parameters n and m.

Convex grid drawings of planar graphs with constant edge-vertex resolution

In this work, we continue the study of the area required for convex straight-line grid drawings of 3-connected plane graphs, which has been intensively investigated in the last decades. Motivated by applications, such as graph editors, we additionally require the obtained drawings to have bounded edge-vertex resolution, that is, the closest distance between a vertex and any non-incident edge in the drawing is lower bounded by a constant that does not depend on the size of the graph. We present a drawing algorithm that takes as input a 3-connected plane graph with n vertices and f internal faces, and computes a convex straight-line drawing with edge-vertex resolution at least 12 on an integer grid of size (n−2+a)×(n−2+a), where a=min⁡{n−3,f}. Our result improves the previously best-known area bound of (3n−7)×(3n−7)/2 by Chrobak, Goodrich and Tamassia.

Path eccentricity of graphs

Let G be a connected graph. The eccentricity of a path P, denoted by eccG(P), is the maximum distance from P to any vertex in G. In the Central path (CP) problem, our aim is to find a path of minimum eccentricity. This problem was introduced by Cockayne et al., in 1981, in the study of different centrality measures on graphs. They showed that CP can be solved in linear time in trees, but it is known to be NP-hard in many classes of graphs such as chordal bipartite graphs, planar 3-connected graphs, split graphs, etc. We investigate the path eccentricity of a connected graph G as a parameter. Let pe(G) denote the value of eccG(P) for a central path P of G. We obtain tight upper bounds for pe(G) in some graph classes. We show that pe(G)≤1 on biconvex graphs and design an algorithm that finds such a path in linear time. On the other hand, by investigating the longest paths of a graph, we obtain tight upper bounds for pe(G) on arbitrary graphs and k-connected graphs. Finally, we study the relation between a central path and a longest path in a graph. We show that, on bipartite permutation graphs, a longest path is also a central path. Furthermore, for any such class of graphs, we exhibit a superclass that does not satisfy this property.

Independence numbers of polyhedral graphs

A polyhedral graph is a 3-connected planar graph. We find the least possible order p(k,a) of a polyhedral graph containing a k-independent set of size a for all positive integers k and a. In the case k=1 and a even, we prove that the extremal graphs are exactly the vertex-face (radial) graphs of maximal planar graphs.

The Lower Bipartite Number of a Graph

For a graph G, we define the lower bipartite number LB(G) as the minimum order of a maximal induced bipartite subgraph of G. We study the parameter, and the related parameter bipartite domination, providing bounds both in general graphs and in some graph families. For example, we show that there are arbitrarily large 4-connected planar graphs G with LB(G) = 4 but a 5-connected planar graph has linear LB(G). We also show that if G is a maximal outerplanar graph of order n, then LB(G) lies between (n + 2)/3 and 2 n/3, and these bounds are sharp.

Planar and Toroidal Morphs Made Easier

We present simpler algorithms for two closely related morphing problems, both based on the barycentric interpolation paradigm introduced by Floater and Gotsman, which is in turn based on Floater’s asymmetric extension of Tutte’s classical spring-embedding theorem. First, we give a very simple algorithm to construct piecewise-linear morphs between planar straight-line graphs. Our algorithm entirely avoids the classical edge-collapsing strategy dating back to Cairns; instead, in each morphing step, we interpolate the pair of weights associated with a single edge. Specifically, given equivalent straight-line drawings $\Gamma_0$ and $\Gamma_1$ of the same 3-connected planar graph $G$, with the same convex outer face, we construct a morph from $\Gamma_0$ to $\Gamma_1$ that consists of $O(n)$ unidirectional morphing steps, in $O(n^{1+\omega/2})$ time, where $n$ is the number of vertices and $\omega$ is the matrix multiplication constant. Second, we describe a natural extension of barycentric interpolation to geodesic graphs on the flat torus. Barycentric interpolation cannot be applied directly in this setting, because the linear systems defining intermediate vertex positions are not necessarily solvable. We describe a simple scaling strategy that circumvents this issue. Computing the appropriate scaling requires $O(n^{\omega/2})$ time, after which we can compute the drawing at any point in the morph in $O(n^{\omega/2})$ time. Our algorithm is considerably simpler than the recent algorithm of Chambers, Erickson, Lin, and Parsa, and produces more natural morphs. Our techniques also yield a simple proof of a conjecture of Connelly, Henderson, Ho, and Starbird for geodesic torus triangulations.

Random cubic planar graphs converge to the Brownian sphere

In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney’s theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere.

Constructions of DNA and polypeptide cages based on plane graphs and odd crossing [formula omitted]-junctions

The constructions of three-dimensional synthetic DNA and polypeptide structures with a single closed DNA strand and polypeptide chain are mathematically based on strong traces of polyhedral graphs. However, a method developed for constructing such a DNA and polypeptide structure may impose additional restrictions on the types of strong traces and polyhedral graphs. In this paper, we show that strong traces for certain 2-connected plane graphs (allowed to have multiple edges) can be obtained using thickened graphs (sometimes called ribbon graphs) constructed with only two types of junctions : 0-crossing junction and special d(v)-crossing junction (called π-junction), where d(v) is the degree of the vertex v at which the d(v)-crossing junction is to be placed. The π-junctions are only applicable to vertices with odd degrees (≥3). We characterize the 2-connected plane graphs to which our approach can be applied and provide a brief guideline for the implementation of our method. This approach provides the theory, as well as a set of candidates, for designing and constructing stable DNA and polypeptide molecules needing only a method capable of creating the 0-crossing and π-junctions in a 2-connected plane graph.

Enumeration of labelled 4-regular planar graphs II: Asymptotics

This work is a follow-up of the article (Noy et al., 2019), where the authors solved the problem of counting labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic estimate for the number gn of labelled 4-regular planar graphs on n vertices. Our estimate is of the form gn∼g⋅n−7/2ρ−nn!, where g>0 is a constant and ρ≈0.24377 is the radius of convergence of the generating function ∑n≥0gnxn/n!, and conforms to the universal pattern obtained previously in the enumeration of several classes of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to deal with large systems of multivariate polynomial equations. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.

On the Hamiltonian Property Hierarchy of 3-Connected Planar Graphs

The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. The graph $G$ is prism-hamiltonian if the prism over $G$ has a Hamilton cycle. A good even cactus is a connected graph in which every block is either an edge or an even cycle and every vertex is contained in at most two blocks. It is known that good even cacti are prism-hamiltonian. Indeed, showing the existence of a spanning good even cactus has become the most common technique in proving prism-hamiltonicity. Špacapan [S. Špacapan. A counterexample to prism-hamiltonicity of 3-connected planar graphs. J. Combin. Theory Ser. B, 146:364--371, 2021] asked whether having a spanning good even cactus is equivalent to having a hamiltonian prism for 3-connected planar graphs. In this article we answer his question in the negative, by showing that there are infinitely many 3-connected planar prism-hamiltonian graphs that have no spanning good even cactus. In addition, we prove the existence of an infinite class of 3-connected planar graphs that have a spanning good even cactus but no spanning good even cactus with maximum degree three.

Distribution of subtree sums

Motivated by the conjectures of Bondy and Malkevitch on (almost-)pancyclicity of 4-connected planar graphs, we study the distribution of subtree weights in trees with positive integer vertex weights. Let T be a tree on N1 vertices and let c:V(T)→N be vertex weights such that ∑v∈V(T)c(v)=N2. Let TS(T;c) denote the set of weights of subtrees of T, and [N1,K]N≔{k∈N:N1≤k≤K}. We show that, for every integer N1≤K≤N2, if N2<2N1−1, we have |TS(T;c)∩[N1,K]N|≥K−N1+12. As a corollary, we show that, for every planar hamiltonian graph G with minimum degree at least 4 and for every integer K with N≤K≤|V(G)|, where N≔⌈|V(G)|2⌉+3, there exist K−N+12 distinct cycle lengths in [N,K]N.