An O(n+m) Certifying Triconnnectivity Algorithm for Hamiltonian Graphs (original) (raw)

A simple algorithm for triconnectivity of a multigraph

2009

Vertex-connectivity and edge-connectivity represent the extent to which a graph is connected. Study of these key properties of graphs plays an important role in varieties of computer science applications. Recent years have witnessed a number of linear time 3-edge-connectivity algorithms - with increasing simplicity. In contrast, the state-of-the-art algorithm for 3-vertex-connectivity due to Hopcroft and Tarjan lacks the simplicity in the sense of ease of implementation as well as the number of passes over the graph although its time and space complexity is theoretically linear. In this paper, we propose a linear time reduction from 3-vertex-connectivity to 3-edge-connectivity of a multigraph. This reduction was previously unknown, while the reduction in the opposite direction already exists. We apply an existing linear time 3-edge-connectivity algorithm on the reduced graph for solving the 3-vertex-connectivity of the original graph. Hence, for a graph with |V| vertices and |E| edg...

Deciding Graph non-Hamiltonicity via a Closure Algorithm

international symposium on algorithms and computation, 2016

We present a matching and LP based heuristic algorithm that decides graph non-Hamiltonicity. Each of the n! Hamilton cycles in a complete directed graph on n + 1 vertices corresponds with each of the n! n-permutation matrices P , such that p u,i = 1 if and only if the i th arc in a cycle enters vertex u, starting and ending at vertex n + 1. A graph instance (G) is initially coded as exclusion set E, whose members are pairs of components of P , {p u,i , p v,i+1 }, i = 1, n − 1, for each arc (u, v) not in G. For each {p u,i , p v,i+1 } ∈ E, the set of P satisfying p u,i = p v,i+1 = 1 correspond with a set of cycles not in G. Accounting for all arcs not in G, E codes precisely the set of cycles not in G. A doubly stochastic-like O(n 4) formulation of the Hamilton cycle decision problem is then constructed. Each {p u,i , p v,j } is coded as variable q u,i,v,j such that the set of integer extrema is the set of all permutations. We model G by setting each q u,i,v,j = 0 in correspondence with each {p u,i , p v,j } ∈ E such that for non-Hamiltonian G, integer solutions cannot exist. We then recognize non-Hamiltonicity by iteratively deducing additional q u,i,v,j that can be set zero and expanding E until the formulation becomes infeasible, in which case we recognize that no integer solutions exists i.e. G is decided non-Hamiltonian. The algorithm first chooses any {p u,i , p v,j } ∈ E and sets q u,i,v,j = 1. As a relaxed LP, if the formulation is infeasible, we deduce q u,i,v,j = 0 and {p u,i , p v,j } can be added to E. Then we choose another {p u,i , p v,j } ∈ E and start over. Otherwise, as a subset of matching problems together with a subset of necessary conditions, if q u,i,v,j cannot participate in a match, we deduce q u,i,v,j = 0 and {p u,i , p v,j } can be added E. We again choose another {p u,i , p v,j } ∈ E and start over. Otherwise q u,i,v,j is undecided, and we exhaustively test all {p u,i , p v,j } ∈ E. If E becomes the set of all {p u,i , p v,j }, G is decided non-Hamiltonian. Otherwise G is undecided. We call this the Weak Closure Algorithm. Only non-Hamiltonian G share this maximal property. Over 100 non-Hamiltonian graphs (10 through 104 vertices) and 2000 randomized 31 vertex non-Hamiltonian graphs are tested and correctly decided non-Hamiltonian. For Hamiltonian G, the complement of E provides information about covers of matchings, perhaps useful in searching for cycles. We also present an example where the WCA fails to deduce any integral value for any q u,i,v,j i.e. G is undecided.

Certifying Fully Dynamic Algorithms for Recognition and Hamiltonicity of Threshold and Chain Graphs

Algorithmica

Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for example connectivity, shortest paths and graph recognition. In this paper we present fully dynamic algorithms for the recognition of threshold graphs and chain graphs, which are optimal in the sense that the costs per modification are linear in the number of modified edges. Furthermore, our algorithms also consider the addition and deletion of sets of vertices as well as edges. In the negative case, i.e., where the graph is not a threshold graph or chain graph anymore, our algorithms return a certificate of constant size. Additionally, we present optimal fully dynamic algorithms for the Hamiltonian cycle problem and the Hamiltonian path problem on threshold and chain graphs which return a ver...

An efficient certifying algorithm for the Hamiltonian cycle problem on circular-arc graphs

Theoretical Computer Science, 2011

A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is an evidence that can be used to authenticate the correctness of the answer. A Hamiltonian cycle in a graph is a simple cycle in which each vertex of the graph appears exactly once. The Hamiltonian cycle problem is to determine whether or not a graph contains a Hamiltonian cycle. The best result for the Hamiltonian cycle problem on circular-arc graphs is an O(n 2 log n)-time algorithm, where n is the number of vertices of the input graph. In fact, the O(n 2 log n)-time algorithm can be modified as a certifying algorithm although it was published before the term certifying algorithms appeared in the literature. However, whether there exists an algorithm whose time complexity is better than O(n 2 log n) for solving the Hamiltonian cycle problem on circular-arc graphs has been opened for two decades. In this paper, we present an O(∆n)time certifying algorithm to solve this problem, where ∆ represents the maximum degree of the input graph. The certificates provided by our algorithm can be authenticated in O(n) time.

Why Depth-First Search Efficiently Identifies Two and Three-Connected Graphs

Lecture Notes in Computer Science, 2010

Given an undirected 3-connected graph G with n vertices and m edges, we modify depth-first search to produce a sparse spanning subgraph with at most 4n − 10 edges that is still 3-connected. If G is 2-connected, to maintain 2-connectivity, the resulting graph will have at most 2n − 3 edges. The way depth-first search discards irrelevant edges illustrates the reason behind its ability to verify and certify biconnectivity [1-3] and triconnectivity in linear time. Dealing with a sparser graph, after the first depth-first-search calls, makes the algorithms in [2, 5] more efficient. We also give a characterization of separation pairs of a 2-connected graph in terms of the resulting sparse graph. *

A distributed biconnectivity check

2006

For many distributed autonomous robotic systems, it is important to maintain communication connectivity among the robots. That is, each robot must be able to communicate with each other robot, perhaps through a series of other robots. Ideally, this property should be robust to the removal of any single robot from the system. In this work, we define a property of a team's communication graph that ensures this property, called biconnectivity.

Sufficient Condition and Algorithm for Hamiltonian in 3-Connected 3-Regular Planar Bipartite Graph

International Journal of Computer Applications (ISSN: 0975 – 8887), 2015

A graph G (V, E) is said to be Hamiltonian if it contains a spanning cycle. The spanning cycle is called a Hamiltonian cycle of G and G is said to be a Hamiltonian graph. A Hamiltonian path is a path that contains all the vertices in V (G) but does not return to the vertex in which it began. In this paper, we study Hamiltonicity of 3-connected, 3-regular planar bipartite graph G with partite sets V=M  N. We shall prove that G has a Hamiltonian cycle if G is balanced with M = N. For that we present an algorithm for a bipartite graph KM,N where M>3, N>3 and M,N both are even to possess a Hamiltonian cycle. In particular, we also prove a theorem for S proper subset (M or N) of V the number of components W (G-S) = S implies the graph has a Hamiltonian path.

Proving hamiltonian properties in connected 4-regular graphs: an ILP-based approach

2021

In this paper we study some open questions related to the smallest order f(C,¬H) of a 4-regular graph which has a connectivity property C but does not have a hamiltonian property H. In particular, C is either connectivity, 2-connectivity or 1-toughness and H is hamiltonicity, homogeneous traceability or traceability. A standard theoretical approach to these questions had already been used in the literature, but in many cases did not succeed in determining the exact value of f(). Here we have chosen to use Integer Linear Programming and to encode the graphs that we are looking for as the binary solutions to a suitable set of linear inequalities. This way, there would exist a graph of order n with certain properties if and only if the corresponding ILP had a feasible solution, which we have determined through a branch-and-cut procedure. By using our approach, we have been able to compute f(C,¬H) for all the pairs of considered properties with the exception of C =1-toughness, H =tracea...

Distributed 2-vertex connectivity test of graphs using local knowledge

In Proceeding of the International …, 2007

Abstract—The vertex connectivity of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. This work is devoted to the problem of vertex connectivity test of graphs in a distributed environment based on a general and a ...

Connectivity, Traceability and Hamiltonicity

La Matematica, 2022

Let G be a simple, connected, triangle-free graph with minimum degree δ, order n and leaf number L(G). If G has a cut-vertex, we prove that L(G) ≥ 4δ − 4 and n ≥ 4δ − 1. Both lower bounds are sharp. The lower bound on the leaf number strengthens a result by Mukwembi for triangle-free graphs. As corollaries, we deduce sufficient conditions for connectivity, traceability and Hamiltonicity in triangle-free graphs. As an easy extension of a result by Goodman and Hedetiniemi, we show that a simple, connected, claw-free, paw-free graph G is Hamiltonian if and only if G is not a path. We consider only simple graphs, that is, graphs with neither loops nor multiple edges.