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 Cycle transversals in bounded degree graphs

2011 ◽  
Vol Vol. 13 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Marina Groshaus ◽  
Pavol Hell ◽  
Sulamita Klein ◽  
Loana Tito Nogueira ◽  
Fábio Protti
Keyword(s):  
Approximation Algorithms ◽  
Polynomial Time ◽  
Maximum Degree ◽  
Decomposition Theorem ◽  
Np Hard ◽  
Input Graph ◽  
Bounded Degree ◽  
Time Approximation ◽  
International Audience ◽  
Bounded Degree Graphs

Graphs and Algorithms International audience In this work we investigate the algorithmic complexity of computing a minimum C(k)-transversal, i.e., a subset of vertices that intersects all the chordless cycles with k vertices of the input graph, for a fixed k \textgreater= 3. For graphs of maximum degree at most three, we prove that this problem is polynomial-time solvable when k \textless= 4, and NP-hard otherwise. For graphs of maximum degree at most four, we prove that this problem is NP-hard for any fixed k \textgreater= 3. We also discuss polynomial-time approximation algorithms for computing C(3)-transversals in graphs of maximum degree at most four, based on a new decomposition theorem for such graphs that leads to useful reduction rules.

scholarly journals Linear choosability of graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Louis Esperet ◽  
Mickael Montassier ◽  
André Raspaud
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Oriented Graph ◽  
Vertex Coloring ◽  
Proper Coloring ◽  
Maximum Average Degree ◽  
Complete Problem ◽  
International Audience ◽  
Np Complete ◽  
Proper Vertex

International audience A proper vertex coloring of a non oriented graph $G=(V,E)$ is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph $G$ is $L$-list colorable if for a given list assignment $L=\{L(v): v∈V\}$, there exists a proper coloring $c$ of $G$ such that $c(v)∈L(v)$ for all $v∈V$. If $G$ is $L$-list colorable for every list assignment with $|L(v)|≥k$ for all $v∈V$, then $G$ is said $k$-choosable. A graph is said to be lineary $k$-choosable if the coloring obtained is linear. In this paper, we investigate the linear choosability of graphs for some families of graphs: graphs with small maximum degree, with given maximum average degree, planar graphs... Moreover, we prove that determining whether a bipartite subcubic planar graph is lineary 3-colorable is an NP-complete problem.

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 Consecutive Colouring of Oriented Graphs

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Marta Borowiecka-Olszewska ◽  
Ewa Drgas-Burchardt ◽  
Nahid Yelene Javier-Nol ◽  
Rita Zuazua
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Oriented Graphs ◽  
Complete Problem ◽  
Np Complete

AbstractWe consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

EFFICIENT APPROACH FOR WEB SERVICES SELECTION WITH MULTI-QoS CONSTRAINTS

2008 ◽  
Vol 17 (03) ◽  
pp. 349-371 ◽  
Author(s):  
TAO HUANG ◽  
LEI LI ◽  
JUN WEI
Keyword(s):  
Web Services ◽  
Polynomial Time ◽  
High Performance ◽  
Feasible Solution ◽  
Search Space ◽  
User Preference ◽  
Composite Service ◽  
Efficient Approach ◽  
Complete Problem ◽  
Np Complete

With the increasing number of Web Services with similar or identical functionality, the non-functional properties of a Web Service will become more and more important. Hence, a choice needs to be made to determine which services are to participate in a given composite service. In general, multi-QoS constrained Web Services composition, with or without optimization, is a NP-complete problem on computational complexity that cannot be exactly solved in polynomial time. A lot of heuristics and approximation algorithms with polynomial- and pseudo-polynomial-time complexities have been designed to deal with this problem. However, these approaches suffer from excessive computational complexities that cannot be used for service composition in runtime. In this paper, we propose a efficient approach for multi-QoS constrained Web Services selection. Firstly, a user preference model was proposed to collect the user's preference. And then, a correlation model of candidate services are established in order to reduce the search space. Based on these two model, a heuristic algorithm is then proposed to find a feasible solution for multi-QoS constrained Web Services selection with high performance and high precision. The experimental results show that the proposed approach can achieve the expecting goal.

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 The resolving number of a graph Delia

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Delia Garijo ◽  
Antonio González ◽  
Alberto Márquez
Keyword(s):  
Graph Theory ◽  
Polynomial Time ◽  
Maximum Degree ◽  
Clique Number ◽  
Metric Dimension ◽  
Np Hard ◽  
Arbitrary Graph ◽  
International Audience ◽  
Graph Parameters ◽  
Two Parameters

Graph Theory International audience We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the metric dimension of an arbitrary graph is known to be NP-hard, we show that the resolving number can be computed in polynomial time. We then relate the resolving number to classical graph parameters: diameter, girth, clique number, order and maximum degree. With these relations in hand, we characterize the graphs with resolving number 3 extending other studies that provide characterizations for smaller resolving number.

scholarly journals Polynomial time solution to NP-complete problem Hamiltonian cycle (P=NP Proof)

2021 ◽  
Author(s):  
Yasaman KalantarMotamedi
Keyword(s):  
Computer Science ◽  
Polynomial Time ◽  
Hamiltonian Cycle ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Rigorous Proof ◽  
Complete Problem ◽  
Time Solution ◽  
Np Complete

P vs NP is one of the open and most important mathematics/computer science questions that has not been answered since it was raised in 1971 despite its importance and a quest for a solution since 2000. P vs NP is a class of problems that no polynomial time algorithm exists for any. If any of the problems in the class gets solved in polynomial time, all can be solved as the problems are translatable to each other. One of the famous problems of this kind is Hamiltonian cycle. Here we propose a polynomial time algorithm with rigorous proof that it always finds a solution if there exists one. It is expected that this solution would address all problems in the class and have a major impact in diverse fields including computer science, engineering, biology, and cryptography.

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 Graphs with no induced wheel and no induced antiwheel

2015 ◽  
Vol 9 (2) ◽  
pp. 357-366 ◽  
Author(s):  
Frédéric Maffray
Keyword(s):  
Polynomial Time ◽  
Simple Structure ◽  
Complete Problem ◽  
Complementary Graph ◽  
Np Complete ◽  
Chordless Cycle

A wheel is a graph that consists of a chordless cycle of length at least 4 plus a vertex with at least three neighbors on the cycle. An antiwheel is the complementary graph of a wheel. It was shown recently that detecting induced wheels is an NP-complete problem. In contrast, it is shown here that graphs that contain no wheel and no antiwheel have a very simple structure and consequently can be recognized in polynomial time.

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