scholarly journals On paths, trails and closed trails in edge-colored graphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Laurent Gourvès ◽  
Adria Lyra ◽  
Carlos A. Martinhon ◽  
Jérôme Monnot
Keyword(s):  
Polynomial Time ◽  
Optimal Solution ◽  
Colored Graph ◽  
Edge Disjoint ◽  
Colored Graphs ◽  
International Audience ◽  
Np Complete ◽  
T Path ◽  
Vertex Disjoint

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

scholarly journals Some algorithmic results for finding compatible spanning circuits in edge-colored graphs

2020 ◽  
Vol 40 (4) ◽  
pp. 1008-1019
Author(s):  
Zhiwei Guo ◽  
Hajo Broersma ◽  
Ruonan Li ◽  
Shenggui Zhang
Keyword(s):  
Polynomial Time ◽  
Sufficient Conditions ◽  
Complete Graphs ◽  
Hamilton Cycles ◽  
Colored Graph ◽  
Colored Graphs ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Np Complete ◽  
Euler Tours

Abstract A compatible spanning circuit in a (not necessarily properly) edge-colored graph G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. Sufficient conditions for the existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and Euler tours), and polynomial-time algorithms for finding compatible Euler tours have been considered in previous literature. More recently, sufficient conditions for the existence of more general compatible spanning circuits in specific edge-colored graphs have been established. In this paper, we consider the existence of (more general) compatible spanning circuits from an algorithmic perspective. We first show that determining whether an edge-colored connected graph contains a compatible spanning circuit is an NP-complete problem. Next, we describe two polynomial-time algorithms for finding compatible spanning circuits in edge-colored complete graphs. These results in some sense give partial support to a conjecture on the existence of compatible Hamilton cycles in edge-colored complete graphs due to Bollobás and Erdős from the 1970s.

scholarly journals The extended equivalence and equation solvability problems for groups

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Gabor Horvath ◽  
Csaba Szabo
Keyword(s):  
Polynomial Time ◽  
Equivalence Problem ◽  
Nilpotent Groups ◽  
60Th Birthday ◽  
Special Issue ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Solvability Problem ◽  
Np Complete ◽  
Equation Solvability

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience We prove that the extended equivalence problem is solvable in polynomial time for finite nilpotent groups, and coNP-complete, otherwise. We prove that the extended equation solvability problem is solvable in polynomial time for finite nilpotent groups, and NP-complete, otherwise.

scholarly journals Acyclic colourings of graphs with bounded degree

2010 ◽  
Vol Vol. 12 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Mieczyslaw Borowiecki ◽  
Anna Fiedorowicz ◽  
Katarzyna Jesse-Jozefczyk ◽  
Elzbieta Sidorowicz
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Cubic Graph ◽  
Bounded Degree ◽  
Complete Problem ◽  
International Audience ◽  
Np Complete

Graphs and Algorithms International audience A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgraph induced in G by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, there are no bichromatic alternating cycles. In 1999 Boiron et al. conjectured that a graph G with maximum degree at most 3 has an acyclic 2-colouring such that the set of vertices in each colour induces a subgraph with maximum degree at most 2. In this paper we prove this conjecture and show that such a colouring of a cubic graph can be determined in polynomial time. We also prove that it is an NP-complete problem to decide if a graph with maximum degree 4 has the above mentioned colouring.

scholarly journals Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

2021 ◽  
Author(s):  
Thomas Bläsius ◽  
Philipp Fischbeck ◽  
Tobias Friedrich ◽  
Maximilian Katzmann
Keyword(s):  
Computational Complexity ◽  
Structural Properties ◽  
Random Graphs ◽  
Polynomial Time ◽  
Real World ◽  
Degree Distribution ◽  
High Probability ◽  
Vertex Cover ◽  
Optimal Solution ◽  
Np Complete

AbstractThe computational complexity of the VertexCover problem has been studied extensively. Most notably, it is NP-complete to find an optimal solution and typically NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice.

scholarly journals Solving equations over small unary algebras

2005 ◽  
Vol DMTCS Proceedings vol. AF,... (Proceedings) ◽  
Author(s):  
Przemyslaw Broniek
Keyword(s):  
Polynomial Time ◽  
Partial Characterization ◽  
Polynomial Equations ◽  
Solving Equations ◽  
International Audience ◽  
Np Complete ◽  
System Of Polynomial Equations

International audience We consider the problem of solving a system of polynomial equations over fixed algebra $A$ which we call MPolSat($A$). We restrict ourselves to unary algebras and give a partial characterization of complexity of MPolSat($A$). We isolate a preorder $P(A)$ to show that when $A$ has at most 3 elements then MPolSat($A$) is in $P$ when width of $P(A)$ is at most 2 and is NP-complete otherwise. We show also that if $P ≠ NP$ then the class of unary algebras solvable in polynomial time is not closed under homomorphic images.

scholarly journals Edge-Disjoint Branchings in Temporal Digraphs

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Victor Campos ◽  
Raul Lopes ◽  
Andrea Marino ◽  
Ana Silva
Keyword(s):  
Polynomial Time ◽  
Classical Result ◽  
The Other ◽  
Static Case ◽  
Edge Disjoint ◽  
Other Hand ◽  
Time Stamps ◽  
Np Complete

A temporal digraph ${\cal G}$ is a triple $(G, \gamma, \lambda)$ where $G$ is a digraph, $\gamma$ is a function on $V(G)$ that tells us the time stamps when a vertex is active, and $\lambda$ is a function on $E(G)$ that tells for each $uv\in E(G)$ when $u$ and $v$ are linked. Given a static digraph $G$, and a subset $R\subseteq V(G)$, a spanning branching with root $R$ is a subdigraph of $G$ that has exactly one path from $R$ to each $v\in V(G)$. In this paper, we consider the temporal version of Edmonds' classical result about the problem of finding $k$ edge-disjoint spanning branchings respectively rooted in given $R_1,\cdots,R_k$. We introduce and investigate different definitions of spanning branchings, and of edge-disjointness in the context of temporal digraphs. A branching ${\cal B}$ is vertex-spanning if the root is able to reach each vertex $v$ of $G$ at some time where $v$ is active, while it is temporal-spanning if each $v$ can be reached from the root at every time where $v$ is active. On the other hand, two branchings ${\cal B}_1$ and ${\cal B}_2$ are edge-disjoint if they do not use the same edge of $G$, and are temporal-edge-disjoint if they can use the same edge of $G$ but at different times. This lead us to four definitions of disjoint spanning branchings and we prove that, unlike the static case, only one of these can be computed in polynomial time, namely the temporal-edge-disjoint temporal-spanning branchings problem, while the other versions are $\mathsf{NP}$-complete, even under very strict assumptions. 

scholarly journals P6- and triangle-free graphs revisited: structure and bounded clique-width

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Andreas Brandstädt ◽  
Tilo Klembt ◽  
Suhail Mahfud
Keyword(s):  
Polynomial Time ◽  
Dominating Set ◽  
Vertex Cover ◽  
Maximum Clique ◽  
Stable Set ◽  
Induced Matching ◽  
International Audience ◽  
Np Complete ◽  
Maximum Induced Matching ◽  
Second Order Logic

International audience The Maximum Weight Stable Set (MWS) Problem is one of the fundamental problems on graphs. It is well-known to be NP-complete for triangle-free graphs, and Mosca has shown that it is solvable in polynomial time when restricted to P6- and triangle-free graphs. We give a complete structure analysis of (nonbipartite) P6- and triangle-free graphs which are prime in the sense of modular decomposition. It turns out that the structure of these graphs is extremely simple implying bounded clique-width and thus, efficient algorithms exist for all problems expressible in terms of Monadic Second Order Logic with quantification only over vertex predicates. The problems Vertex Cover, MWS, Maximum Clique, Minimum Dominating Set, Steiner Tree, and Maximum Induced Matching are among them. Our results improve the previous one on the MWS problem by Mosca with respect to structure and time bound but also extends a previous result by Fouquet, Giakoumakis and Vanherpe which have shown that bipartite P6-free graphs have bounded clique-width. Moreover, it covers a result by Randerath, Schiermeyer and Tewes on polynomial time 3-colorability of P6- and triangle-free graphs.

scholarly journals Complexity of conditional colouring with given template

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Peter J. Dukes ◽  
Steve Lowdon ◽  
Gary Macgillivray
Keyword(s):  
Graph Theory ◽  
Polynomial Time ◽  
Vertex Set ◽  
International Audience ◽  
Partitioning Problem ◽  
Np Complete ◽  
Graph Families ◽  
General Collection

Graph Theory International audience We study partitions of the vertex set of a given graph into cells that each induce a subgraph in a given family, and for which edges can have ends in different cells only when those cells correspond to adjacent vertices of a fixed template graph H. For triangle-free templates, a general collection of graph families for which the partitioning problem can be solved in polynomial time is described. For templates with a triangle, the problem is in some cases shown to be NP-complete.

scholarly journals An algorithmic analysis of Flood-It and Free-Flood-It on graph powers

2014 ◽  
Vol Vol. 16 no. 3 (Analysis of Algorithms) ◽  
Author(s):  
Uéverton dos Santos Souza ◽  
Fábio Protti ◽  
Maise Silva
Keyword(s):  
Polynomial Time ◽  
Analysis Of Algorithms ◽  
Np Hard ◽  
Colored Graph ◽  
Combinatorial Game ◽  
Polynomial Time Algorithms ◽  
Minimum Number ◽  
International Audience ◽  
Algorithmic Analysis

Analysis of Algorithms International audience Flood-it is a combinatorial game played on a colored graph G whose aim is to make the graph monochromatic using the minimum number of flooding moves, relatively to a fixed pivot. Free-Flood-it is a variant where the pivot can be freely chosen for each move of the game. The standard versions of Flood-it and Free-Flood-it are played on m ×n grids. In this paper we analyze the behavior of these games when played on other classes of graphs, such as d-boards, powers of cycles and circular grids. We describe polynomial time algorithms to play Flood-it on C2n (the second power of a cycle on n vertices), 2 ×n circular grids, and some types of d-boards (grids with a monochromatic column). We also show that Free-Flood-it is NP-hard on C2n and 2 ×n circular grids.

scholarly journals Connected Tropical Subgraphs in Vertex-Colored Graphs

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Jean-Alexandre Anglès d'Auriac ◽  
Nathann Cohen ◽  
Hakim El Mafthoui ◽  
Ararat Harutyunyan ◽  
Sylvain Legay ◽  
...  
Keyword(s):  
Random Graph ◽  
Upper Bounds ◽  
Interval Graphs ◽  
Np Hard ◽  
Colored Graph ◽  
Colored Graphs ◽  
International Audience ◽  
Split Graphs

International audience A subgraph of a vertex-colored graph is said to be tropical whenever it contains each color of the graph. In this work we study the problem of finding a minimal connected tropical subgraph. We first show that this problem is NP-Hard for trees, interval graphs and split graphs, but polynomial when the number of colors is logarithmic in terms of the order of the graph (i.e. FPT). We then provide upper bounds for the order of the minimal connected tropical subgraph under various conditions. We finally study the problem of finding a connected tropical subgraph in a randomly vertex-colored random graph.

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