Graph Withn Vertices Research Articles

Exact Distance Oracles for Planar Graphs with Failing Vertices

We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex u , a target vertex v and a set X of k failed vertices, such an oracle returns the length of a shortest u -to- v path that avoids all vertices in X . We propose oracles that can handle any number k of failures. We show several tradeoffs between space, query time, and preprocessing time. In particular, for a directed weighted planar graph with n vertices and any constant k , we show an Õ( n )-size, Õ(√ n )-query-time oracle. 1 We then present a space vs. query time tradeoff: for any q ε [ 1,√ n ], we propose an oracle of size n k+1+o(1) / q 2k that answers queries in Õ( q ) time. For single vertex failures ( k = 1), our n 2+o(1) / q 2 -size, Õ( q )-query-time oracle improves over the previously best known tradeoff of Baswana et al. SODA 2012 by polynomial factors for q ≥ n t , for any t ∈ (0,1/2]. For multiple failures, no planarity exploiting results were previously known.

The Characterizing Properties of (Signless) Laplacian Permanental Polynomials of Almost Complete Graphs

Let G be a graph with n vertices, and let L G and Q G denote the Laplacian matrix and signless Laplacian matrix, respectively. The Laplacian (respectively, signless Laplacian) permanental polynomial of G is defined as the permanent of the characteristic matrix of L G (respectively, Q G ). In this paper, we show that almost complete graphs are determined by their (signless) Laplacian permanental polynomials.

On the Circumference of 3-Connected Cubic Triangle-Free Plane Graphs

The circumference of a graph G is the length of a longest cycle in G , denoted by cir G . For any even number n , let c n = min { cir G | G is a 3-connected cubic triangle-free plane graph with n vertices}. In this paper, we show that an upper bound of c n is n + 1 − 3 ⌊ n / 136 ⌋ for n ≥ 136 .

Parallel Minimum Cuts in Near-linear Work and Low Depth

We present the first near-linear work and poly-logarithmic depth algorithm for computing a minimum cut in an undirected graph. Previous parallel algorithms with poly-logarithmic depth required at least quadratic work in the number of vertices. In a graph with n vertices and m edges, our randomized algorithm computes the minimum cut with high probability in O ( m log 4 n ) work and O (log 3 n ) depth. This result is obtained by parallelizing a data structure that aggregates weights along paths in a tree, in addition exploiting the connection between minimum cuts and approximate maximum packings of spanning trees. In addition, our algorithm improves upon bounds on the number of cache misses incurred to compute a minimum cut.

The Minimum Merrifield–Simmons Index of Unicyclic Graphs with Diameter at Most Four

The Merrifield–Simmons index i G of a graph G is defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, i.e., the number of independent vertex sets of G . In this paper, we determine the minimum Merrifield–Simmons index of unicyclic graphs with n vertices and diameter at most four.

Aα-Spectral Characterizations of Some Joins

LetGbe a graph withnvertices. For every realα∈0,1, writeAαGfor the matrixAαG=αDG+1−αAG, whereAGandDGdenote the adjacency matrix and the degree matrix ofG, respectively. The collection of eigenvalues ofAαGtogether with multiplicities are called theAα-spectrum ofG. A graphGis said to be determined by itsAα-spectrum if all graphs having the sameAα-spectrum asGare isomorphic toG. In this paper, we show that some joins are determined by theirAα-spectra forα∈0,1/2 or 1/2,1.

Tight Bounds for Undirected Graph Exploration with Pebbles and Multiple Agents

We study the problem of deterministically exploring an undirected and initially unknown graph with n vertices either by a single agent equipped with a set of pebbles or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles Θ(log n ) bits of memory are necessary and sufficient to explore any graph with at most n vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory Θ(log log n ) pebbles are necessary and sufficient for exploration. We further prove that using collaborating agents instead of pebbles does not help as Θ(log log n ) agents with constant memory each are necessary and sufficient for exploration. For the upper bounds, we devise an algorithm for a single agent with constant memory that explores any n -vertex graph using O (log log n ) pebbles, even when n is not known a priori . The algorithm terminates after polynomial time and returns to the starting vertex. We further show that the algorithm can be realized with additional constant-memory agents rather than pebbles, implying that O (log log n ) agents with constant memory can explore any n -vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph with at most n vertices is already Ω(log log n ) when we allow each agent to have at most O ((log n ) 1 -ε) bits of memory for any ε > 0. Our argument also implies that a single agent with sublogarithmic memory needs Θ(log log n ) pebbles to explore any n -vertex graph.

New Cover Time Bounds for the Coalescing-Branching Random Walk on Graphs

We present new bounds on the cover time of the coalescing-branching random walk process COBRA. The COBRA process, introduced in Dutta et al. [9], can be viewed as spreading a single item of information throughout an undirected graph in synchronised rounds. In each round, each vertex that has received the information in the previous round (possibly simultaneously from more than one neighbour and possibly not for the first time), “pushes” the information to k randomly selected neighbours. The COBRA process is typically studied for integer branching rates k ≥ 2 (with the case k =1 corresponding to a random walk). The aim of the process is to propagate the information quickly, but with a limited number of transmissions per vertex per round. The COBRA cover time is the expected number of rounds until all vertices have received the information at least once. Our main results are bounds of O ( m + ( d max ) 2 log n ) and O ( m log n ) on the COBRA cover time for arbitrary connected graphs with n vertices, m edges and maximum graph degree d max , and bounds of O (( r 2 + r /(1−λ)) log n ) and O ((1/(1− λ) 2 )log n ) for r -regular connected graphs with the second largest eigenvalue λ in absolute value. Our bounds for general graphs are always O ( n 2 log n ), decreasing to O ( n ) for constant degree graphs, while the best previous bound is O ( n 2.75 log n ). Our first bound for regular graphs applied to the lazy variant of the COBRA process is O (( r 2 + r /φ 2 )log n ), where φ is the conductance of the graph. The best previous results for the COBRA process imply for this case only a bound of O (( r 4 /φ 2 )log 2 n ). To derive our bounds, we develop the following new approach to analysing the performance of the COBRA process. We introduce a type of infection process, which we call the Biased Infection with Persistent Source (BIPS) process, show that BIPS can be viewed as dual to COBRA, and obtain bounds for COBRA by analysing the convergence of BIPS.

Windrose Planarity

Given a planar graph G and a partition of the neighbors of each vertex v in four sets v ↗ , v ↖ , v ↙ , and v ↘ , the problem W indrose P lanarity asks to decide whether G admits a windrose-planar drawing , that is, a planar drawing in which (i) each neighbor u ∈ v ↗ v is above and to the right of v , (ii) each neighbor u ∈ v ↖ is above and to the left of v , (iii) each neighbor u ∈ v ↙ is below and to the left of v , (iv) each neighbor u ∈ v ↘ is below and to the right of v , and (v) edges are represented by curves that are monotone with respect to each axis. By exploiting both the horizontal and the vertical relationship among vertices, windrose-planar drawings allow us to simultaneously visualize two partial orders defined by means of the edges of the graph. Although the problem is NP -hard in the general case, we give a polynomial-time algorithm for testing whether there exists a windrose-planar drawing that respects a given combinatorial embedding. This algorithm is based on a characterization of the plane triangulations admitting a windrose-planar drawing. Furthermore, for any embedded graph with n vertices that has a windrose-planar drawing, we can construct one with at most one bend per edge and with at most 2 n −5 bends in total, which lies on the 3 n × 3 n grid. The latter result contrasts with the fact that straight-line windrose-planar drawings may require exponential area.

Better Bounds for Coalescing-Branching Random Walks

Coalescing-branching random walks, or cobra walks for short, are a natural variant of random walks on graphs that can model the spread of disease through contacts or the spread of information in networks. In a k -cobra walk, at each timestep, a subset of the vertices are active; each active vertex chooses k random neighbors (sampled independently and uniformly with replacement) that become active at the next step, and these are the only active vertices at the next step. A natural quantity to study for cobra walks is the cover time, which corresponds to the expected time when all nodes have become infected or received the disseminated information. In this article, we extend previous results for cobra walks in multiple ways. We show that the cover time for the 2-cobra walk on [0, n ] d is O ( n ) (where the order notation hides constant factors that depend on d ); previous work had shown the cover time was O ( n ⋅polylog( n )). We show that the cover time for a 2-cobra walk on an n -vertex d -regular graph with conductance φ G is O ( d 4 φ −2 G log 2 n ), significantly generalizing a previous result that held only for expander graphs with sufficiently high expansion. And, finally, we show that the cover time for a 2-cobra walk on a graph with n vertices and m edges is always O ( mn 3/4 log n ; this is the first result showing that the bound of Θ( n 3 ) for the worst-case cover time for random walks can be beaten using 2-cobra walks.

Cache-Oblivious Buffer Heap and Cache-Efficient Computation of Shortest Paths in Graphs

We present the buffer heap , a cache-oblivious priority queue that supports Delete-Min , Delete , and a hybrid Insert / Decrease-Key operation in O (1/ B log 2 N / M ) amortized block transfers from main memory, where M and B are the (unknown) cache size and block size, respectively, and N is the number of elements in the queue. We introduce the notion of a slim data structure that captures the situation when only a limited portion of the cache, which we call a slim cache , is available to the data structure to retain data between data structural operations. We show that a buffer heap automatically adapts to such an environment and supports all operations in O (1/λ + 1/ B log 2 N /λ) amortized block transfers each when the size of the slim cache is λ. Our results provide substantial improvements over known trivial cache performance bounds for cache-oblivious priority queues with Decrease-Keys . Using the buffer heap, we present cache-oblivious implementations of Dijkstra’s algorithm for undirected and directed single-source shortest path (SSSP) problems for graphs with non-negative real edge-weights. On a graph with n vertices and m edges, our algorithm for the undirected case performs O ( n + m / B log 2 n / M ) block transfers and for the directed case performs O (( n + m / B ) ċ log 2 n / B ) block transfers. These results give the first non-trivial cache-oblivious bounds for the SSSP problem on general graphs. For the all-pairs shortest path (APSP) problem on weighted undirected graphs, we incorporate slim buffer heaps into multi-buffer-buffer-heaps and use these to improve the cache-aware cache complexity. We also present a simple cache-oblivious APSP algorithm for unweighted undirected graphs that performs O ( mn / B log M / B n / B ) block transfers. This matches the cache-aware bound and is a substantial improvement over the previous cache-oblivious bound for the problem.

Approximating the Diameter of Planar Graphs in Near Linear Time

We present a (1 + ε)-approximation algorithm running in O ( f (ε) · n log 4 n ) time for finding the diameter of an undirected planar graph with n vertices and with nonnegative edge lengths.

A Bounded Budget Network Creation Game

We introduce a network creation game in which each player (vertex) has a fixed budget to establish links to other players. In this model, each link has a unit price, and each agent tries to minimize its cost, which is either its eccentricity or its total distance to other players in the underlying (undirected) graph of the created network. Two versions of the game are studied: In the MAX version, the cost incurred to a vertex is the maximum distance between the vertex and other vertices, and, in the SUM version, the cost incurred to a vertex is the sum of distances between the vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard. We take the social cost of the created network to be its diameter, and next we study the maximum possible diameter of an equilibrium graph with n vertices in various cases. When the sum of players’ budgets is n − 1, the equilibrium graphs are always trees, and we prove that their maximum diameter is Θ( n ) and Θ(log n ) in MAX and SUM versions, respectively. When each vertex has a unit budget (i.e., can establish a link to just one vertex), the diameter of any equilibrium graph in either version is Θ(1). We give examples of equilibrium graphs in the MAX version, such that all vertices have positive budgets and yet the diameter is Ω(√log n ). This interesting (and perhaps counterintuitive) result shows that increasing the budgets may increase the diameter of equilibrium graphs and hence deteriorate the network structure. Then we prove that every equilibrium graph in the SUM version has diameter 2 O (√log n ) . Finally, we show that if the budget of each player is at least k , then every equilibrium graph in the SUM version is k -connected or has a diameter smaller than 4.

On the Diameter of Random Planar Graphs

We show that the diameter diam(Gn) of a random labelled connected planar graph withnvertices is equal ton1/4+o(1), in probability. More precisely, there exists a constantc> 0 such that$$ P(\D(G_n)\in(n^{1/4-\e},n^{1/4+\e}))\geq 1-\exp(-n^{c\e}) $$for ε small enough andn ≥ n0(ε). We prove similar statements for 2-connected and 3-connected planar graphs and maps.

Degree lower bounds of tower-type for approximating formulas with parity quantifiers

Kolaitis and Kopparty have shown that for any first-order formula with parity quantifiers over the language of graphs, there is a family of multivariate polynomials of constant-degree that agree with the formula on all but a 2 −Ω( n ) -fraction of the graphs with n vertices. The proof bounds the degree of the polynomials by a tower of exponentials whose height is the nesting depth of parity quantifiers in the formula. We show that this tower-type dependence is necessary. We build a family of formulas of depth q whose approximating polynomials must have degree bounded from below by a tower of exponentials of height proportional to q . Our proof has two main parts. First, we adapt and extend the results by Kolaitis and Kopparty that describe the joint distribution of the parities of the numbers of copies of small subgraphs in a random graph to the setting of graphs of growing size. Second, we analyze a variant of Karp's graph canonical labeling algorithm and exploit its massive parallelism to get a formula of low depth that defines an almost canonical pre-order on a random graph.

Laplacian Spectral Characterization of Some Unicyclic Graphs

LetW(n;q,m1,m2)be the unicyclic graph withnvertices obtained by attaching two paths of lengthsm1andm2at two adjacent vertices of cycleCq. LetU(n;q,m1,m2,…,ms)be the unicyclic graph withnvertices obtained by attachingspaths of lengthsm1,m2,…,msat the same vertex of cycleCq. In this paper, we prove thatW(n;q,m1,m2)andU(n;q,m1,m2,…,ms)are determined by their Laplacian spectra whenqis even.

Some Properties on Estrada Index of Folded Hypercubes Networks

LetGbe a simple graph withnvertices and letλ1,λ2,…,λnbe the eigenvalues of its adjacency matrix; the Estrada indexEEGof the graphGis defined as the sum of the termseλi, i=1,2,…,n. Then-dimensional folded hypercube networksFQnare an important and attractive variant of then-dimensional hypercube networksQn, which are obtained fromQnby adding an edge between any pair of vertices complementary edges. In this paper, we establish the explicit formulae for calculating the Estrada index of the folded hypercubes networksFQnby deducing the characteristic polynomial of the adjacency matrix in spectral graph theory. Moreover, some lower and upper bounds for the Estrada index of the folded hypercubes networksFQnare proposed.

Optimal Coadapted Coupling for a Random Walk on the Hyper-Complete Graph

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks onZ2dwas considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk onKnd, whereKnis the complete graph withnvertices. Moreover, we show that although this coupling is not maximal for anyn(i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling asn→ ∞.

The Laplacian Spectral Radius of a Class of Unicyclic Graphs

LetC(n,k)be the set of all unicyclic graphs withnvertices and cycle lengthk. For anyU∈C(n,k),Uconsists of the (unique) cycle (sayCk) of lengthkand a certain number of trees attached to the vertices ofCkhaving (in total)n-kedges. If there are at most two trees attached to the vertices ofCk, wherekis even, we identify in the class of unicyclic graphs those graphs whose Laplacian spectral radii are minimal.

The Largest Laplacian Spectral Radius of Unicyclic Graphs with Fixed Diameter

We identify graphs with the maximal Laplacian spectral radius among all unicyclic graphs withnvertices and diameterd.