Large Planar Graphs Research Articles

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.

Graph theory and the analysis of fracture networks

Two-dimensional exposures of fracture networks can be represented as large planar graphs that comprise a series of branches (B) representing the fracture traces and nodes (N) representing their terminations and linkages. The nodes and branches may link to form connected components (K), which may contain fracture-bounded regions (R) or blocks. The proportions of node types provide a basis for characterizing the topology of the network. The average degree <d> relates the number of branches (|B|) and nodes (|N|) and Euler's formula establishes a link between all four elements of the graph with |N| - |B| + |R| - |K| = 0.Treating a set of fractures as a graph returns the focus of description to the underlying relationships between the fractures and, hence, to the network rather that its constitutive elements. Graph theory provides a wide range of applicable theorems and well-tested algorithms that can be used in the analysis of fault and fracture systems. We discuss a range of applications to two-dimensional fracture and fault networks, and briefly discuss application to three-dimensions.

Complex network and fractal theory for the assessment of water distribution network resilience to pipe failures

AbstractWater distribution networks (WDNs) must keep a proper level of service under a wide range of operational conditions, and, in particular, the analysis of their resilience to pipe failures is essential to improve their design and management. WDNs can be regarded as large sparse planar graphs showing fractal and complex network properties. In this paper, the relationship linking the geometrical and topological features of a WDN to its resilience to the failure of a pipe is investigated. Some innovative indices have been borrowed from fractal geometry and complex network theory to study WDNs. Considering all possible network configurations obtained by suppressing one link, the proposed indices are used to quantify the impact of pipe failure on the system's resilience. This approach aims to identify critical links, in terms of resilience, with the help of topological metrics only, and without recourse to hydraulic simulations, which require complex calibration processes and come with a computational burden. It is concluded that the proposed procedure, which has been successfully tested on two real WDNs located in southern Italy, can provide valuable information to water utilities about which pipes have a significant role in network performance, thus helping in their design, planning and management.

Access Time Oracle for Planar Graphs

The study of urban networks reveals that the accessibility of important city objects for the vehicle traffic and pedestrians is significantly correlated to the popularity, micro-criminality, micro-economic vitality, and social liveability of the city, and is always the chief factor in regulating the growth and expansion of the city. The accessibility between different components of an urban structure are frequently measured along the streets and routes considered as edges of a planar graph, while the traffic ultimate destination points and street junctions are treated as vertices. For estimation of the accessibility of destination vertex $j$ from vertex $i$ through urban networks, in particular, the random walks are used to calculate the expected distance a random walker starting from $i$ makes before $j$ is visited (known as access time ). The state-of-the-art of access time computation is costly in large planar graphs since it involves matrix operation over entire graph. The time complexity is $O(n^{2.376})$ where $n$ is the number of vertices in the planar graph. To enable efficient access time query answering in large planar graphs, this work proposes the first access time oracle which is based on the proposed access time decomposition and reconstruction scheme. The oracle is a hierarchical data structure with deliberate design on the relationships between different hierarchical levels. The storage requirement of the proposed oracle is $O(n^{\frac{4}{3}}\log \log n)$ and the access time query response time is $O(n^{\frac{2}{3}})$ . The extensive tests on a number of large real-world road networks (with up to about 2 million vertices) have verified the superiority of the proposed oracle.

On the Fiedler value of large planar graphs

The Fiedler value λ2, also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs G with n vertices, denoted by λ2max, and we show the bounds 2+Θ(1n2)⩽λ2max⩽2+O(1n). We also provide bounds on the maximum Fiedler value for the following classes of planar graphs: Bipartite planar graphs, bipartite planar graphs with minimum vertex-degree 3, and outerplanar graphs. Furthermore, we derive almost tight bounds on λ2max for two more classes of graphs, those of bounded genus and Kh-minor-free graphs.

Sparse graphs of girth at least five are packable

A graph is packable if it is a subgraph of its complement. The following statement was conjectured by Faudree, Rousseau, Schelp and Schuster in 1981: every non-star graph G with girth at least 5 is packable.The conjecture was proved by Faudree et al. with the additional condition that G has at most 65n−2 edges. In this paper, for each integer k≥3, we prove that every non-star graph with girth at least 5 and at most 2k−1kn−αk(n) edges is packable, where αk(n) is o(n) for every k. This implies that the conjecture is true for sufficiently large planar graphs.

On the Fiedler value of large planar graphs (Extended abstract)

Abstract The Fiedler value λ 2 , also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study upper bounds on the Fiedler value of planar graphs. Let λ 2 max be the maximum Fiedler value among all planar graphs G with n vertices. We show here the bounds 2 + Θ ( 1 n 2 ) ⩽ λ 2 max ⩽ 2 + O ( 1 n ) . Similar lower and upper bounds on the maximum Fiedler value are obtained for the classes of bipartite planar graphs, bipartite planar graphs with minimum vertex degree 3 and outerplanar graphs. We also derive almost tight bounds on λ 2 max for the classes of graphs of bounded genus and K h -minor-free graphs. The proofs rely on a result of Spielman and Teng stating that λ 2 ⩽ 8 Δ n for any planar graph with n vertices and maximum vertex degree Δ, on a result of Kelner stating that λ 2 = O ( g n ) for genus g graphs of bounded degree, and on the separator theorem.

3-Connected Cores In Random Planar Graphs

The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints.In this paper we focus on the structure of random 2-connected planar graphs regarding the sizes of their 3-connected building blocks, which we callcores. In fact, we prove a general theorem regarding random biconnected graphs from various classes. IfBnis a graph drawn uniformly at random from a suitable classof labelled biconnected graphs, then we show that with probability 1 −o(1) asn→ ∞,Bnbelongs to exactly one of the following categories:(i)either there is a uniquegiantcore inBn, that is, there is a 0 &lt;c=c() &lt; 1 such that the largest core contains ~cnvertices, and every other core contains at mostnαvertices, where 0 &lt; α = α() &lt; 1;(ii)or all cores ofBncontainO(logn) vertices.Moreover, we find the critical condition that determines the category to whichBnbelongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 andn. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particularc= 0.765. . . and α = 2/3.

Computational study on planar dominating set problem

Recently, there has been significant theoretical progress towards fixed-parameter algorithms for the DOMINATING SET problem of planar graphs. It is known that the problem on a planar graph with n vertices and dominating number k can be solved in O ( 2 O ( k ) n ) time using tree/branch-decomposition based algorithms. In this paper, we report computational results of Fomin and Thilikos algorithm which uses the branch-decomposition based approach. The computational results show that the algorithm can solve the DOMINATING SET problem of large planar graphs in a practical time and memory space for the class of graphs with small branchwidth. For the class of graphs with large branchwidth, the size of instances that can be solved by the algorithm in practice is limited to about one thousand edges due to a memory space bottleneck. The practical performances of the algorithm coincide with the theoretical analysis of the algorithm. The results of this paper suggest that the branch-decomposition based algorithms can be practical for some applications on planar graphs.

Planarity-preserving clustering and embedding for large planar graphs

In this paper we present a novel approach for cluster-based drawing of large planar graphs that maintains planarity. Our technique works for arbitrary planar graphs and produces a clustering which satisfies the conditions for compound-planarity (c-planarity). Using the clustering, we obtain a representation of the graph as a collection of O(log n) layers, where each succeeding layer represents the graph in an increasing level of detail. At the same time, the difference between two graphs on neighboring layers of the hierarchy is small, thus preserving the viewer's mental map. The overall running time of the algorithm is O( nlog n), where n is the number of vertices of graph G.

Large planar graphs with given diameter and maximum degree

We consider the problem of determining the maximum number of vertices in a planar graph with given maximum degree Δ and diameter k. This number has previously been exactly determined when k = 2. We show here that when k = 3, the number is roughly between 4.5Δ and 8Δ. We also show that in general the number is Θ(Δ ⌊ k 2 ⌋) for any fixed value of k.

Heuristics for rapidly four-coloring large planar graphs

We present several algorithms for rapidly four-coloring large planar graphs and discuss the results of extensive experimentation with over 140 graphs from two distinct classes of randomly generated instances having up to 128,000 vertices. Although the algorithms can potentially require exponential time, the observed running times of our more sophisticated algorithms are linear in the number of vertices over the range of sizes tested. The use of Kempe chaining and backtracking together with a fast heuristic which usually, but not always, resolves impasses gives us hybrid algorithms that: (1) successfully four-color all our test graphs, and (2) in practice run, on average, only twice as slow as the well-known, nonexact, simple to code, ź(n) saturation algorithm of Brelaz.

On Cycles and Connectivity in Planar Graphs

Let G be a graph and ζ(G) be the greatest integer n such that every set of n points in G lies on a cycle [8]. It is clear that ζ(G)≥2 for 2-connected planar graphs. Moreover, it is easy to construct arbitrarily large 2-connected planar graphs for which ζ=2. On the other hand, by a well-known theorem of Tutte [5], [6], if G is planar and 4-connected, it has a Hamiltonian cycle, i.e., ζ(G)=|V(G)| for all 4-connected (and hence for all 5-connected) planar graphs.