Unsolved Problems on Distance in Graphs

The distance between a pair of nodes of a graph G is the length of a shortest path connecting them. The eccentricity of a node v is the greatest distance between v and another node. The radius and diameter of a graph are, respectively, the smallest and largest eccentricities among its nodes. The status of v is the sum of the distances from v to all other nodes. We shall discuss various conjectures and unsolved problems concerning distance concepts in graphs. These problems involve radius, diameter, and status, as well as other distance concepts such as distance sequences, domination, distance in digraphs, and convexity.     

On the distance spectral radius of some graphs

For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. In this paper, we determine the unique graph with minimum distance spectral radius among all connected graphs of order n with a given diameter. Moreover, we determine the unique graph with maximum distance spectral radius among the catacondensed hexagonal systems with h hexagons.     

Sphere of influence graphs in general metric spaces

Let χ = {X₁…, X_n} be a set of points in a metric space (n ≥ 2). Let r_i, denote the minimum distance between X_i and the other points in χ. The sphere of influence at X_i is the open ball with center X_iand radius r_i. The sphere of influence graph has vertex set χ with an edge joining a pair of distinct vertices provided the corresponding spheres of influence intersect. This proximity graph was introduced by Toussaint to model computer vision and pattern recognition problems in the Euclidean plane. We discuss the abstract properties of sphere of influence graphs in general metric spaces with emphasis on normed linear spaces. Our work generalizes and simplifies some theorems in the literature on sphere of influence graphs and lays the groundwork for future work.     

Some graft transformations and its applications on the distance spectral radius of a graph

Let D(G)=(d_i,j)_n×n denote the distance matrix of a connected graph G with order n, where d_ij is equal to the distance between v_i and v_j in G. The largest eigenvalue of D(G) is called the distance spectral radius of graph G, denoted by ?(G). In this paper, some graft transformations that decrease or increase ?(G) are given. With them, for the graphs with both order n and k pendant vertices, the extremal graphs with the minimum distance spectral radius are completely characterized; the extremal graph with the maximum distance spectral radius is shown to be a dumbbell graph (obtained by attaching some pendant edges to each pendant vertex of a path respectively) when 2≤k≤n−2; for k=1,2,3,n−1, the extremal graphs with the maximum distance spectral radius are completely characterized.     

A decomposition approach for solving a broadcast domination network design problem

We consider an optimization problem that integrates network design and broadcast domination decisions. Given an undirected graph, a feasible broadcast domination is a set of nonnegative integer powers f _i assigned to each node i, such that for any node j in the graph, there exists some node k having a positive f _k -value whose shortest distance to node j is no more than f _k . The cost of a broadcast domination solution is the sum of all node power values. The network design problem constructs edges that decrease the minimum broadcast domination cost on the graph. The overall problem we consider minimizes the sum of edge construction costs and broadcast domination costs. We show that this problem is NP-hard in the strong sense, even on unweighted graphs. We then propose a decomposition strategy, which iteratively adds valid inequalities based on optimal broadcast domination solutions corresponding to the first-stage network design solutions. We demonstrate that our decomposition approach is computationally far superior to the solution of a single large-scale mixed-integer programming formulation.     

Short containers in Cayley graphs

The star diameter of a graph measures the minimum distance from any source node to several other target nodes in the graph. For a class of Cayley graphs from abelian groups, a good upper bound for their star diameters is given in terms of the usual diameters and the orders of elements in the generating subsets. This bound is tight for several classes of graphs including hypercubes and directed n-dimensional tori. The technique used is the so-called disjoint ordering for a system of subsets, due to Gao, Novick and Qiu [S. Gao, B. Novick, K. Qiu, From Hall’s matching theorem to optimal routing on hypercubes, J. Comb. Theory B 74 (1998) 291-301].     

Optimal gathering protocols on paths under interference constraints

We study the problem of gathering information from the nodes of a multi-hop radio network into a predefined destination node under reachability and interference constraints. In such a network, a node is able to send messages to other nodes within reception distance, but doing so it might create interference with other communications. Thus, a message can only be properly received if the receiver is reachable from the sender and there is no interference from another message being transmitted simultaneously. The network is modeled as a graph, where the vertices represent the nodes of the network and the edges, the possible communications. The interference constraint is modeled by a fixed integer d≥1, which implies that nodes within distance d in the graph from one sender cannot receive messages from another node. In this paper, we suppose that each node has one unit-length message to transmit and, furthermore, we suppose that it takes one unit of time (slot) to transmit a unit-length message and during such a slot we can have only calls which do not interfere (called compatible calls). A set of compatible calls is referred to as a round. We give protocols and lower bounds on the minimum number of rounds for the gathering problem when the network is a path and the destination node is either at one end or at the center of the path. These protocols are shown to be optimal for any d in the first case, and for 1≤d≤4, in the second case.     

Mobile geometric graphs: detection, coverage and percolation

We consider the following dynamic Boolean model introduced by van den Berg et al. (Stoch. Process. Appl. 69:247–257, 1997). At time 0, let the nodes of the graph be a Poisson point process in ${\mathbb{R}^d}$ with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes whose distance is at most r. We study three fundamental problems in this model: detection (the time until a target point—fixed or moving—is within distance r of some node of the graph); coverage (the time until all points inside a finite box are detected by the graph); and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these quantities by combining ideas from stochastic geometry, coupling and multi-scale analysis.     

On a binary distance model for the minimum linear arrangement problem

The minimum linear arrangement problem consists of finding an embedding of the nodes of a graph on the line such that the sum of the resulting edge lengths is minimized. The problem is among the classical NP-hard optimization problems and there has been extensive research on exact and approximative algorithms. In this paper, we introduce a new model based on binary variables d _ijk that are equal to 1 if nodes i and j have distance k in the ordering. We analyze this model and point to connections and differences to a model using integer distance variables. Based on computational experiments, we argue that our model is worth further theoretical and practical investigation and that is has potentials yet to be examined.     

Most unbreakable murky graphs are bull-free

A bull is the (self-complementary) graph with nodesa, b, c, d, e and edgesab, ac, bc, bd,ce. A star cutset in a graph G is a set C of nodes such thatG-C is disconnected, and such that some node inC is adjacent to all remaining nodes inC. A graph is called unbreakable if it has more than two nodes and if neither the graph nor its complement has a star cutset. Hayward defined a graphG to be murky if neitherG nor its complement contains a chordless cycle with five nodes or a chordless path with six nodes. We prove that most unbreakable murky graphs are bull-free. This leads, via a result of Chvátal and Sbihi, to a shortening of Hayward’s proof that murky graphs are perfect.     

On Integral Sum Graphs

A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u)+f(v) = f(w) for some node w in G. A node of G is called a saturated node if it is adjacent to every other node of G. We show that any integral sum graph which is not K₃ has at most two saturated nodes. We determine the structure for all integral sum graphs with exactly two saturated nodes, and give an upper bound for the number of edges of a connected integral sum graph with no saturated nodes. We introduce a method of identification on constructing new connected integral sum graphs from given integral sum graphs with a saturated node. Moreover, we show that every graph is an induced subgraph of a connected integral sum graph. Miscellaneous relative results are also presented.     

On minimum power connectivity problems

Given a (directed or undirected) graph with edge costs, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Motivated by applications for wireless networks, we present polynomial and improved approximation algorithms, as well as inapproximability results, for some classic network design problems under the power minimization criteria. Our main result is for the problem of finding a min-power subgraph that contains k internally-disjoint vs-paths from every node v to a given node s: we give a polynomial algorithm for directed graphs and a logarithmic approximation algorithm for undirected graphs. In contrast, we will show that the corresponding edge-connectivity problems are unlikely to admit even a polylogarithmic approximation.     

On integral sum graphs

A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u)+f(v)=f(w) for some node w in G. A node of G is called a saturated node if it is adjacent to every other node of G. We show that any integral sum graph which is not K₃ has at most two saturated nodes. We determine the structure for all integral sum graphs with exactly two saturated nodes, and give an upper bound for the number of edges of a connected integral sum graph with no saturated nodes. We introduce a method of identification on constructing new connected integral sum graphs from given integral sum graphs with a saturated node. Moreover, we show that every graph is an induced subgraph of a connected integral sum graph. Miscellaneous related results are also presented.     

On the distance spectral radius of cacti

A cactus is a connected graph in which any two cycles have at most one common vertex. In this article, we determine the unique graph with minimal distance spectral radius in the class of all cacti with n vertices and k cycles. Also, we determine the unique graph with minimal distance spectral radius in the class of all cacti with n vertices and r pendent vertices. Moreover, we determine the class of cacti in which the maximal distance spectral radius among all cacti with n vertices and k cycles is attained.     

Hop constrained Steiner trees with multiple root nodes

We consider a network design problem that generalizes the hop and diameter constrained Steiner tree problem as follows: Given an edge-weighted undirected graph with two disjoint subsets representing roots and terminals, find a minimum-weight subtree that spans all the roots and terminals so that the number of hops between each relevant node and an arbitrary root does not exceed a given hop limit H. The set of relevant nodes may be equal to the set of terminals, or to the union of terminals and root nodes. This article proposes integer linear programming models utilizing one layered graph for each root node. Different possibilities to relate solutions on each of the layered graphs as well as additional strengthening inequalities are then discussed. Furthermore, theoretical comparisons between these models and to previously proposed flow- and path-based formulations are given. To solve the problem to optimality, we implement branch-and-cut algorithms for the layered graph formulations. Our computational study shows their clear advantages over previously existing approaches.     

An Integer Linear Programming Formulation and Branch-and-Cut Algorithm for the Capacitated m-Ring-Star Problem

We study the capacitated m-ring-star problem (CmRSP) that faces the design of minimum cost network structure that connects customers with m rings using a set of ring connections that share a distinguished node (depot), and optionally star connections that connect customers to ring nodes. Ring and star connections have some associated costs. Also, rings can include transit nodes, named Steiner nodes, to reduce the total network cost if possible. The number of customers in each ring-star (ringʼs customers and customer connected to it through star connections) have an upper bound (capacity).These kind of networks are appropriate in optical fiber urban environments. CmRSP is know to be NP-Hard. In this paper we propose an integer linear programming formulation and a branch-and-cut algorithm.     

A Characterization of the Hamming Graphs and the Dual Polar Graphs by Completely Regular Subgraphs

In this paper we study a distance-regular graph Γ of diameter d ≥? 3 which satisfies the following two conditions: (a) For any integer i with 1 ≤? i ≤? d ? 1 and for any pair of vertices at distance i in Γ there exists a strongly closed subgraph of diameter i containing them; (b) There exists a strongly closed subgraph Δ which is completely regular in Γ. It is known that if Δ has diameter 1, then Γ is a regular near polygon. We prove that if a strongly closed subgraph Δ of diameter j with 2 ≤? j ≤? d ? 1 is completely regular of covering radius d ? j in Γ, then Γ is either a Hamming graph or a dual polar graph.     

The complexity of node blocking for dags

We consider the following modification of annihilation games called node blocking. Given a directed graph, each vertex can be occupied by at most one token. There are two types of tokens, each player can move only tokens of his type. The players alternate their moves and the current player i selects one token of type i and moves the token along a directed edge to an unoccupied vertex. If a player cannot make a move then he loses. We consider the problem of determining the complexity of the game: given an arbitrary configuration of tokens in a planar directed acyclic graph (dag), does the current player have a winning strategy? We prove that the problem is PSPACE-complete.     

Broadcasting in one dimension

We consider the broadcasting problem for one-dimensional grid graphs with a given neighborhood template. There are two different models that have been considered-shouting (a node informs all of its neighbors in one step) and whispering (a node informs a single neighbor in one step). Let σ(t) (respectively ω(t)) denote the maximum number of nodes that can be reached in t steps by shouting (respectively whispering) broadcast from a single source.We obtain detailed information about the benefits of shouting over whispering. We prove for the one-dimensional case a conjecture by Stout that ω(t) eventually becomes a polynomial. In particular, we show that there exist constants i and t₀ such that ω(t)=σ(t)−i for all t ≥ t₀. When the broadcast only goes in one direction (i.e., when all elements of the template are positive), we also determine that i=d −1 and t₀≤3d for a neighborhood template with the furthest neighbor at distance d.