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The eccentricity of a node in a graph is defined as the length of a longest shortest path starting at that node. The eccentricity distribution over all nodes is a relevant descriptive property of the graph, and its extreme values allow the derivation of measures such as the radius, diameter, center and periphery of the graph. This paper describes two new methods for computing the eccentricity distribution of large graphs such as social networks, web graphs, biological networks and routing networks.We first propose an exact algorithm based on eccentricity lower and upper bounds, which achieves significant speedups compared to the straightforward algorithm when computing both the extreme values of the distribution as well as the eccentricity distribution as a whole. The second algorithm that we describe is a hybrid strategy that combines the exact approach with an efficient sampling technique in order to obtain an even larger speedup on the computation of the entire eccentricity distribution. We perform an extensive set of experiments on a number of large graphs in order to measure and compare the performance of our algorithms, and demonstrate how we can efficiently compute the eccentricity distribution of various large real-world graphs. [ABSTRACT FROM AUTHOR]

Metacyclic graphs, which include supertoroids as a subclass, have been shown to possess interesting properties and potential applications in implementing moderate- to large-size parallel processors with fairly small node degrees. Wu, Lakshmivarahan, and Dhall (J. Parallel Distrib. Comput. 60 (2000), pp. 539-565) have described a deterministic, distributed routing scheme for certain subclasses of metacyclic graphs. However, they offer no proof that the scheme is a shortest-path routing algorithm and do not indicate whether or how their scheme may be extended to the entire class of metacyclic graphs. In this paper, we provide a near-shortest-path, deterministic routing algorithm that is applicable to any metacyclic graph and derive a bound for the diameter of such graphs. [ABSTRACT FROM AUTHOR]

A star is a graph in which some node is incident with every edge of the graph, i.e., a graph of diameter at most 2. A star forest is a graph in which each connected component is a star. Given a connected graph G in which the edges may be weighted positively. A spanning star forest of G is a subgraph of G which is a star forest spanning the nodes of G. The size of a spanning star forest F of G is defined to be the number of edges of F if G is unweighted and the total weight of all edges of F if G is weighted. We are interested in the problem of finding a Maximum Weight spanning Star Forest (MWSFP) in G. In [C. T. Nguyen, J. Shen, M. Hou, L. Sheng, W. Miller and L. Zhang, Approximating the spanning star forest problem and its applications to genomic sequence alignment, SIAM J. Comput. 38(3) (2008) 946-962], the authors introduced the MWSFP and proved its NP-hardness. They also gave a polynomial time algorithm for the MWSF problem when G is a tree. In this paper, we present a linear time algorithm that solves the MSWF problem when G is a cactus. [ABSTRACT FROM AUTHOR]

Abstract Let P be a path graph of n vertices embedded in a metric space. We consider the problem of adding a new edge to P such that the diameter of the resulting graph is minimized. Previously (in ICALP 2015) the problem was solved in O (n log 3 ⁡ n) time. In this paper, based on new observations and different algorithmic techniques, we present an O (n log ⁡ n) time algorithm. [ABSTRACT FROM AUTHOR]

Let γ pr ( G) denote the paired domination number of graph G. A graph G with no isolated vertex is paired domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γ pr ( G – v) < γ pr ( G). We call these graphs γ pr -critical. In this paper, we present a method of constructing γ pr -critical graphs from smaller ones. Moreover, we show that the diameter of a γ pr -critical graph is at most $$\frac{3}{2}(\gamma_{pr} (G)-2)$$ and the upper bound is sharp, which answers a question proposed by Henning and Mynhardt [The diameter of paired-domination vertex critical graphs, Czechoslovak Math. J., to appear]. [ABSTRACT FROM AUTHOR]

Knödel graphs Wd,n are regular graphs on n vertices and degree d. They have been introduced by W. Knödel and have been proved to be minimum gossip graphs and minimum broadcast graphs for d = ⌊log n⌋. They became even more interesting in the light of recent results regarding the diameter, which is, up to now, the smallest known diameter among all minimum broadcast graphs on 2d vertices. Also, the logarithmic time routing algorithm that we have found, the bipancyclicity property, embedding properties and, nevertheless, Cayley graph structure, impel these graphs as good candidates for regular network constructions, especially in supercomputing. In this paper we describe an efficient way to compute the spectra of Knödel graphs using results from Fourier analysis, circulant matrices and PD-matrices. Based on this result we give a formula for the number of spanning trees in Knödel graphs. [ABSTRACT FROM AUTHOR]

In this study, we consider the problem of estimating the diameter of a graph, that is, the maximum distance between any two vertices, in linear time. We address a question posed in the literature-whether there exists an interesting graph class with arbitrarily large cycles for which breadth first search (BFS) would always return high-eccentricity vertices. We answer this question positively, defining a class of graphs that generalizes AT-free graphs and has no bound on the size of induced cycles, yet BFS always returns a vertex whose eccentricity is within a constant difference from the diameter. In addition, we consider the question-also explicitly stated in the literature-whether some variant of the so-called multisweep algorithm would always return a high-eccentricity vertex. We show that the answer is negative by describing a family of graphs for which no variant of multisweep BFS can return a vertex of eccentricity higher than half of the diameter plus a constant. [ABSTRACT FROM AUTHOR]

We provide the first sparse covers and probabilistic partitions for graphs excluding a fixed minor that have strong diameter bounds; i.e. each set of the cover/partition has a small diameter as an induced sub-graph. Using these results we provide improved distributed name-independent routing schemes. Specifically, given a graph excluding a minor on r vertices and a parameter ρ>0 we obtain the following results: (1) a polynomial algorithm that constructs a set of clusters such that each cluster has a strong-diameter of O( r ρ) and each vertex belongs to 2 r! clusters; (2) a name-independent routing scheme with a stretch of O( r), headers of O(log n+ rlog r) bits, and tables of size 2 r! log n/log log n bits; (3) a randomized algorithm that partitions the graph such that each cluster has strong-diameter O( r6 ρ) and the probability an edge ( u, v) is cut is O( r d( u, v)/ ρ). [ABSTRACT FROM AUTHOR]

We present approximation algorithms for two closely related bicriteria problems. Given a graph with two weight functions on the edges, length and cost, we consider the Bounded-Diameter Minimum-Cost Steiner Tree (BDMST) problem and the Bounded-Diameter Minimum-Cardinality Edge Addition (BDMC) problem. We present a parameterized approximation algorithm for the BDMST problem with a bicriteria approximation ratio of (O(p log s/log p),O(log s/log p)) where the first factor gives the relaxation on the diameter bound, the second factor is the cost-approximation factor, s is the number of required nodes and p, 1 ≤ p < s, is an input parameter. The parameter p allows a trade-off between the two approximation factors. This is the first improvement of the cost-approximation factor since the scheme proposed by Marathe et al. [9]. For example, p can be set to sα to obtain an (O(sα),O(1)) approximation for a constant α. The algorithm is very simple and is suitable for distributed implementations. We also present the first algorithm for Bounded-Hops Minimum-Cost Steiner Tree for complete graphs with triangle inequality. This model includes graphs defined by points in a Euclidean space of any dimension. The algorithm guarantees an approximation ratio of (O(logds),O(logds)) where d is the bound on the diameter. This is an improvement over the general-case approximation when d is comparable with s. For example, the ratio is (O(1),O(1)) for any d = sα where α is a constant between 0 and 1. For the case where the number of terminals is a constant and all edge lengths are unit, we have a polynomial-time algorithm. This can be extended to any length function providing a (1 + ε) in the approximation with ε showing up in the time complexity of the algorithm. For another special case, where the cost of any edge is either 1 or 0 and the length of each edge is positive, an algorithm with approximation ratio of (O(log(c log s)), O(log(c log s))) is presented, where c is the cost of the optimal solution. This approximation is a significant improvement over (O(log s),O(log s)) when the cost of the solution c is much smaller than the number of terminals s. This is useful when an existing multicast network is to be modified to accommodate new terminals with the addition of as few new edges as possible. We also propose an approximation algorithm for the Bounded-Diameter Minimum-Cardinality Edge Addition problem, which achieves an O(log n) approximation while relaxing the diameter bound by 2. While this ratio is the same as the one claimed in [3], this algorithm is simple and combinatorial rather than based on the Linear Program solution and can be readily modified for a distributed implementation. [ABSTRACT FROM AUTHOR]