Splitting Plane Graphs to Outerplanarity

On The Number of Subgraphs of Prescribed Type of Planar Graphs With A Given Number of Vertices

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

A maximal planar graph is a simple planar graph in which every face is a triangle. We show here that such graphs with maximum degree Δ and diameter two have no more than 3/2Δ + 1 vertices. We also show that there exist maximal planar graphs with diameter two and exactly [3/2Δ + 1] vertices.

The graph theoretic facility layout problem (GTFLP) seeks the best spatial arrangement of a set of facilities, which minimizes the total transportation cost of material flow between facilities, or maximizes total facility adjacency benefits. The GTFLP has applications in arranging rooms within a building floorplan, placing machines on a factory floor, controls on an instrument panel, or components on a circuit board. For this reason, the GTFLP is an important problem within industrial design. The GTFLP comprises two phases: the generation of a highly weighted maximal planar graph (MPG), whose edges specify adjacencies required in the layout, and the drawing of an orthogonal geometric dual to this MPG, called the block plan, under given area requirements for each facility. The first phase is NP-complete, and has been researched extensively; in the absence of a polynomial time algorithm, it appears there is little future for further research in this area. The second phase however has received far less attention due to the difficulties in not only determining algorithms for generating layouts, but also the hurdles encountered in automating these methods. This paper presents a new approach to finding a block plan from an arbitrary MPG, with facility area requirements.

In this paper, we investigate the problem of graph list colouring in the on-line setting. We provide several results on paintability of graphs in the model introduced by Schauz [18] and Zhu [25]. We prove that the on-line version of Ohba’s conjecture is true for the class of planar graphs. We show that the conjecture for partial list colouring on-line holds for several graph classes, namely claw-free graphs, maximal planar graphs, series-parallel graphs, and chordal graphs.

A subset of vertics W and a planer graph G is a face-independent vertex cover (FIVC) if there exists a plane embedding of G such that each boundary contains exactly one vertex from W. In general, verifying if a plane graph admits a VIFC is an NP-complete problem. If G is a maximal planar graph, then the existence of a FIVC in G can be efficiently tested by using the so-called t-paths between odd-degree vertics of G (a t-path is a sequence of adjacent triangles). Also for Halin Graphs (i.e., skirted trees), its FIVC’s can be generated in polynomial time.

A graph G on n vertices is called non-universal if its maximum degree is at most $$n-2$$ . In this paper, we give a structural characterization for non-universal maximal planar graphs with diameter two. In precise, we find 10 basic graphs, and then generate all 25 non-universal maximal planar graphs with diameter two by adding repeatedly and appropriately 3-vertices to some of these 10 basic graphs. As an application, we show that maximal planar graphs with diameter two are pancyclic except five special graphs.

The article introduces four concepts: maximal simple planar graphs (mspgs), disc graphs, irreducible graphs, and k-irreducible graphs, along with illustrative examples for each.A complete list of graphs that are both mspgs and regular is provided.The paper presents the following results: A maximal simple planar graph with n vertices has 3(n-2) edges and 2(n-2) faces; every simple planar graph contains a vertex with degree 5 or less; and the disc graph D m,k has 2m + 3(k -1) edges and m + 2k -1 faces.The paper establishes the equivalence between the following statements through logical proofs and rigorous analysis: All planar graphs are 4-colourable; all maximal simple planar graphs are 4-colourable; all irreducible graphs are 4-colourable; and all 5-irreducible graphs are 4-colourable.The core result of the paper is: "Every planar graph is 4-colourable if and only if every 5-irreducible graph is 4-colourable."

This paper is devoted to study the combinatorial properties of Local Minimum Spanning Trees (LMSTs), a geometric structure that is attracting increasing research interest in the wireless sensor networks community. Namely, we study which topologies are allowed for a sensor network that uses, for supporting connectivity, a local minimum spanning tree approach. First, we refine the current definition of LMST realizability, focusing on the role of the power of transmission (i.e., of the radius of the covered area). Second, we show simple planar, connected, and triangle-free graphs with maximum degree 3 that cannot be represented as an LMST. Third, we present several families of graphs that can be represented as LMSTs. Then, we show a relationship between planar graphs and their representability as LMSTs based on homeomorphism. Finally, we show that the general problem of determining whether a graph is LMST representable is NP-hard.

A triangulation of a point set in the plane is a maximal planar straight-line graph with the vertex set .In the Minimum Weight Triangulation (MWT) problem, we want to draw a triangulation of a given point set that minimizes the sum of the edges length. Recently, Mulzer and Rote have proved that this problem is NP-hard. In this paper, we present an evolutionary algorithm using Ant Colony Optimization to solve this problem.

Faster Shortest-Path Algorithms for Planar Graphs

An injective colouring of a graph is a colouring in which every two vertices sharing a common neighbour receive a different colour. Chen, Hahn, Raspaud, and Wang conjectured that every planar graph of maximum degree Δ ≥ 3 admits an injective colouring with at most ⌊3Δ/2⌋ colours. This was later disproved by Lužar and Škrekovski for certain small and even values of Δ and they proposed a new refined conjecture. Using an algorithm for determining the injective chromatic number of a graph, i.e. the smallest number of colours for which the graph admits an injective colouring, we give computational evidence for Lužar and Škrekovski’s conjecture and extend their results by presenting an infinite family of 3-connected planar graphs for each Δ (except for 4) attaining their bound, whereas they only gave finitely many examples for each Δ. Hence, together with another infinite family of maximum degree 4, we provide infinitely many counterexamples to the conjecture by Chen et al. for each Δ if 4 ≤ Δ ≤ 7 and every even Δ ≥ 8. We provide similar evidence for analogous conjectures by La and Štorgel and Lužar, Škrekovski, and Tancer when the girth is restricted as well. Also in these cases we provide infinite families of 3-connected planar graphs attaining the bounds of these conjectures for certain maximum degrees Δ ≥ 3.

String graphs of k-bend paths on a grid

Motivated by applications in distributed storage, the storage capacity of a graph was recently defined to be the maximum amount of information that can be stored across the vertices of a graph such that the information at any vertex can be recovered from the information stored at the neighboring vertices. Computing the storage capacity is a fundamental problem in network coding and is related, or equivalent, to some well-studied problems such as index coding with side information and generalized guessing games. In this paper, we consider storage capacity as a natural information-theoretic analogue of the minimum vertex cover of a graph. Indeed, while it was known that storage capacity is upper bounded by minimum vertex cover, we show that by treating it as such we can get a 3/2 approximation for planar graphs, and a 4/3 approximation for triangle-free planar graphs. Since the storage capacity is intimately related to the index coding rate, we get a 2 approximation of index coding rate for planar graphs and 3/2 approximation for triangle-free planar graphs. Previously, only a trivial 4 approximation of the index coding rate was known for planar graphs. We also show a polynomial time approximation scheme for the index coding rate when the alphabet size is constant. We then develop a general method of “gadget covering” to upper bound the storage capacity in terms of the average of a set of vertex covers. This method is intuitive and leads to the exact characterization of storage capacity for various families of graphs. As an illustrative example, we use this approach to derive the exact storage capacity of cycles-with-chords, a family of graphs related to outerplanar graphs. Finally, we generalize the storage capacity notion to include recovery from partial node failures in distributed storage. We show tight upper and lower bounds on this partial recovery capacity that scales nicely with the fraction of failures in a vertex.

Symmetry, such as structural symmetry, color symmetry and so on, plays an important role in graph coloring. In this paper, we use structural symmetry and color symmetry to study the characterization for the neighbor-distinguishing index of planar graphs. Let G be a simple graph with no isolated edges. The neighbor-distinguishing edge coloring of G is a proper edge coloring of G such that any two adjacent vertices admit different sets consisting of the colors of their incident edges. The neighbor-distinguishing index χa′(G) of G is the smallest number of colors in such an edge coloring of G. It was conjectured that if G is a connected graph with at least three vertices and G≠C5, then χa′(G)≤Δ+2. In this paper, we show that if G is a planar graph with maximum degree Δ≥13, then Δ≤χa′(G)≤Δ+1, and, further, χa′(G)=Δ+1 if and only if G contains two adjacent vertices of maximum degree.