Algorithms to count paths and cycles

Simple Undirected Two-Commodity Integral Flow with a Unitary Demand

This work presents a log-space reduction which compresses an $n$-vertex directed a cyclic graph with $m(n)$ sources embedded on a surface of genus $g(n)$, to a graph with $O(m(n)+g(n))$ vertices while preserving reach ability between a given pair of vertices. Applying existing algorithms to this reduced graph yields new deterministic algorithms with improved space bounds as well as improved simultaneous time-space bounds for the reach ability problem over a large class of directed a cyclic graphs. Specifically, it significantly extends the class of surface-embedded graphs with log-space reach ability algorithms: from planar graphs with $O(\log n)$ sources, to graphs with $2^{O(\sqrt{\log n})}$ sources embedded on a surface of genus $2^{O(\sqrt{\log n})}$. Additionally, it yields an $O(n^{1-\epsilon})$ space algorithm with polynomial running time for reach ability over graphs with $O(n^{1-\epsilon})$ sources embedded on surfaces of genus $O(n^{1-\epsilon})$.

We study random graph models for directed acyclic graphs, a class of networks that includes citation networks, food webs, and feed-forward neural networks among others. We propose two specific models roughly analogous to the fixed edge number and fixed edge probability variants of traditional undirected random graphs. We calculate a number of properties of these models, including particularly the probability of connection between a given pair of vertices, and compare the results with real-world acyclic network data finding that theory and measurements agree surprisingly well-far better than the often poor agreement of other random graph models with their corresponding real-world networks.

Centrality measures are the tools used in most of the network analysis to rank the vertices of the network. Depending on different perspectives, many important centrality measures are defined and used for network analysis. Stress is one such important centrality measure of graphs applicable to the study of social and biological networks. In this chapter, we explore the theory describing various properties of the stress centrality measure. Since the adjacency matrix plays an important role in finding the number of shortest paths between a given pair of vertices, an alternate expression to find the stress of a vertex using the adjacency matrix of the graph is presented in this chapter.

Stochastic processes, estimation, and control: The entropy approach: By George N. Saridis

A digraph is semicomplete if it has no pair of non-adjacent vertices. It is complete if every pair of distinct vertices induces a 2-cycle. A digraph is semicomplete multipartite if it can be obtained from a semicomplete digraph D by choosing a collection of vertex-disjoint subsets X₁,…,X_c of V(D) and then deleting all arcs both of whose end-vertices lie inside some X_i. We can also think of a semicomplete digraph as being obtained from a semicomplete multipartite digraph on the same vertex set and partite sets V₁,…,V_c by adding the arcs of a semicomplete digraph D_i on V_i for each partite set V_i. It is well known that both the hamiltonian path and the hamiltonian cycle problem can be solved in polynomial time for semicomplete multipartite digraphs. In this paper we study the complexity of finding a hamiltonian path or cycle in a semicomplete digraph S which is obtained as above from a semicomplete multipartite digraph D and semicomplete digraphs D_i=(V_i,A_i), i∈[c] such that the path or cycle uses as few arcs of A₁∪…A_c as possible. We obtain a number of results for the case when each D_i is a complete digraph. Already this case is highly nontrivial in the cycle case and the complexity is still open. We show how to find a Hamiltonian path which uses as few arcs from the D_i’s as possible in polynomial time and obtain a number of results, both structural and algorithmic on hamiltonian cycles that use the minimum or close to the minimum number of arcs from the D_i’s. Our results imply the polynomial solvability of some special cases of the NP-complete {0,1}-TSP problem. Finally we show that two natural questions about properties of quasi-hamiltonian cycles, that is, cycles meeting all partite sets in semicomplete multipartite digraphs are NP-complete.

Even and odd pairs in comparability and in P4-comparability graphs

Computing a Minimum Weightk-Link Path in Graphs with the Concave Monge Property

Graphs have been widely used to model complex data in many real-world applications. Answering vertex join queries over large graphs is meaningful and interesting, which can benefit friend recommendation in social networks and link prediction, and so on. In this article, we adopt “SimRank” [13] to evaluate the similarity between two vertices in a large graph because of its generality. Note that “Simank” is purely structure dependent, and it does not rely on the domain knowledge. Specifically, we define a S im R ank-based j oin ( SRJ ) query to find all vertex pairs satisfying the threshold from two sets of vertices U and V . To reduce the search space, we propose a shortest-path-distance-based upper bound for SimRank scores to prune unpromising vertex pairs. In the verification, we propose a novel index, called h-go cover + , to efficiently compute the SimRank score of any single vertex pair. Given a graph G , we only materialize the SimRank scores of a small proportion of vertex pairs (i.e., the h-go cover + vertex pairs), based on which the SimRank score of any vertex pair can be computed easily. To find the h-go cover + vertex pairs, we propose an efficient method without building the vertex-pair graph. Hence, large graphs can be dealt with easily. Extensive experiments over both real and synthetic datasets confirm the efficiency of our solution.

Finding a Hamiltonian cycle in a graph G = (V, E) is a well-known problem. The challenge of finding a Hamiltonian cycle that avoids these faults when faulty vertices or edges are present has been extensively studied. When the edge set of G is partitioned into k dimensions, the problem of dimension-balanced Hamiltonian cycles arises, where the Hamiltonian cycle uses approximately the same number of edges from each dimension (differing by at most one). This paper studies whether a dimension-balanced Hamiltonian cycle (DBH) exists in toroidal mesh graphs Tm,n when a single vertex or edge is faulty, called the one-fault DBH problem. We establish that Tm,n is one-fault DBH, except in the following cases: (1) both m and n are even; (2) one of m and n is 3, while the other satisfies mod 4 = 3 and is greater than 6; (3) one of m and n is odd, while the other satisfies mod 4 = 2. Additionally, this paper resolves a conjecture from prior literature, thereby providing a complete solution to the DBP problem on Tm,n.

On the mutually independent Hamiltonian cycles in faulty hypercubes

Hamiltonian cycle problems play a central role in graph theory and have wide-ranging applications in network-on-chip architectures, interconnection networks, and large-scale parallel systems. When a network contains faulty nodes or faulty links, the feasibility of certain paths becomes restricted, making the construction of Hamiltonian cycles substantially more difficult and increasingly important for ensuring reliable communication. A dimension-balanced Hamiltonian cycle is a special type of cycle that maintains an even distribution of edges across multiple dimensions of a network. Its directed counterpart extends this idea to symmetric directed networks by balancing the number of edges used in each positive and negative direction. Such cycles are desirable because they support uniform traffic distribution and reduce communication contention in practical systems. Previous research has examined the existence of directed dimension-balanced Hamiltonian cycles in directed toroidal mesh networks and has shown that some configurations permit directed dimension-balanced Hamiltonian cycles while others do not. Building on this foundation, this paper investigates the fault-tolerant properties of such networks by analyzing whether directed dimension-balanced Hamiltonian cycles still exist when a single vertex (node) or a single edge (link) is faulty. Our results extend the current understanding of Hamiltonian robustness in symmetric directed networks.

Let G=(V,E) be an n-vertices m-edges directed graph. Let s inV be any designated source vertex. We address the problem of reporting the reachability information from s under two vertex failures. We show that it is possible to compute in polynomial time an O(n) size data structure that for any query vertex v, and any pair of failed vertices f_1, f_2, answers in O(1) time whether or not there exists a path from s to v in G\{f_1,f_2}. For the simpler case of single vertex failure such a data structure can be obtained using the dominator-tree from the celebrated work of Lengauer and Tarjan [TOPLAS 1979, Vol. 1]. However, no efficient data structure was known in the past for handling more than one failures. We, in addition, also present a labeling scheme with O(log^3(n))-bit size labels such that for any f_1, f_2, v in V , it is possible to determine in poly-logarithmic time if v is reachable from s in G\{f_1,f_2} using only the labels of f1, f_2 and v. Our data structure can also be seen as an efficient mechanism for verifying double-dominators. For any given x, y, v in V we can determine in O(1) time if the pair (x,y) is a double-dominator of v. Earlier the best known method for this problem was using dominator chain from which verification of double-dominators of only a single vertex was possible.

The pessimistic diagnosability of alternating group graphs under the PMC model

Some properties of [formula omitted]-trees