scholarly journals Proper gap-labellings: on the edge and vertex variants

2019 ◽  
Author(s):  
Celso A. Weffort-Santos ◽  
Christiane N. Campos ◽  
Rafael C. S. Schouery
Keyword(s):  
Computational Complexity ◽  
Simple Graph ◽  
Decision Problems ◽  
Np Completeness ◽  
Polynomial Time Algorithms ◽  
The Family ◽  
Completeness Result ◽  
Np Complete ◽  
Proper Vertex ◽  
Vertex Colouring

Given a simple graph G, an ordered pair (π, cπ) is said to be a gap- [k]-edge-labelling (resp. gap-[k]-vertex-labelling) ofG ifπ is an edge-labelling (vertex-labelling) on the set {1, . . . , k}, and cπ is a proper vertex-colouring such that every vertex of degree at least two has its colour induced by the largest difference among the labels of its incident edges (neighbours). The decision problems associated with these labellings are NP-complete for k ≥ 3, and even when k = 2 for some classes of graphs. This thesis presents a study of the computational complexity of these problems, structural properties for certain families of graphs and several labelling algorithms and techniques. First, we present an NP-completeness result for the family of subcubic bipartite graphs. Second, we present polynomial-time algorithms for families ofgraphs. Third, we introduce a new parameter associated with gap-[k]-vertex-labellings ofgraphs.

scholarly journals 3-Colorability of Pseudo-Triangulations

2015 ◽  
Vol 25 (04) ◽  
pp. 283-298
Author(s):  
Oswin Aichholzer ◽  
Franz Aurenhammer ◽  
Thomas Hackl ◽  
Clemens Huemer ◽  
Alexander Pilz ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Plane Graphs ◽  
Np Completeness ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Complexity Results ◽  
The Family ◽  
General Plane ◽  
Np Complete

Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.

scholarly journals NP-Completeness in the gossip monoid

2018 ◽  
Vol 28 (04) ◽  
pp. 653-672 ◽  
Author(s):  
Peter Fenner ◽  
Marianne Johnson ◽  
Mark Kambites
Keyword(s):  
Algebraic Model ◽  
Decision Problems ◽  
Equivalence Relations ◽  
Membership Problem ◽  
Np Completeness ◽  
Important Decision ◽  
Pure Mathematics ◽  
Finite Set ◽  
Np Complete ◽  
Shed Light

Gossip monoids form an algebraic model of networks with exclusive, transient connections in which nodes, when they form a connection, exchange all known information. They also arise naturally in pure mathematics, as the monoids generated by the set of all equivalence relations on a given finite set under relational composition. We prove that a number of important decision problems for these monoids (including the membership problem, and hence the problem of deciding whether a given state of knowledge can arise in a network of the kind under consideration) are NP-complete. As well as being of interest in their own right, these results shed light on the apparent difficulty of establishing the cardinalities of the gossip monoids: a problem which has attracted some attention in the last few years.

scholarly journals The complexity of deciding whether a graph admits an orientation with fixed weak diameter

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Julien Bensmail ◽  
Romaric Duvignau ◽  
Sergey Kirgizov
Keyword(s):  
Undirected Graph ◽  
Oriented Graph ◽  
Extremal Graph ◽  
Np Completeness ◽  
Directed Paths ◽  
International Audience ◽  
Np Complete ◽  
Distance Properties ◽  
Vertex Colouring

International audience An oriented graph $\overrightarrow{G}$ is said weak (resp. strong) if, for every pair $\{ u,v \}$ of vertices of $\overrightarrow{G}$, there are directed paths joining $u$ and $v$ in either direction (resp. both directions). In case, for every pair of vertices, some of these directed paths have length at most $k$, we call $\overrightarrow{G}$ $k$-weak (resp. $k$-strong). We consider several problems asking whether an undirected graph $G$ admits orientations satisfying some connectivity and distance properties. As a main result, we show that deciding whether $G$ admits a $k$-weak orientation is NP-complete for every $k \geq 2$. This notably implies the NP-completeness of several problems asking whether $G$ is an extremal graph (in terms of needed colours) for some vertex-colouring problems.

scholarly journals Computing some role assignments of Cartesian Product of Graphs

2021 ◽  
Author(s):  
Diane Castonguay ◽  
Elisângela Silva Dias ◽  
Fernanda Neiva Mesquita ◽  
Julliano Rosa Nascimento
Keyword(s):  
Social Networks ◽  
Computational Complexity ◽  
Decision Problem ◽  
Cartesian Product ◽  
Simple Graph ◽  
Graph Models ◽  
Role Assignment ◽  
Cartesian Product Of Graphs ◽  
Np Complete ◽  
Product Of Graphs

In social networks, a role assignment is such that individuals play the same role, if they relate in the same way to other individuals playing counterpart roles. As a simple graph models a social network role assignment rises to the decision problem called r -Role Assignment whether it exists such an assignment of r distinct roles to the vertices of the graph. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is one of the most studied operation on graphs and has numerous applications in diverse areas, such as Mathematics, Computer Science, Chemistry and Biology. In this paper, we determine the computational complexity of r -Role Assignment restricted to Cartesian product of graphs, for r = 2,3. In fact, we show that the Cartesian product of graphs is always 2-role assignable, however the problem of 3-Role Assignment is still NP-complete for this class.

Complexity, computational

2018 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Propositional Logic ◽  
Common Type ◽  
Combinatorial Games ◽  
Np Completeness ◽  
Complete Problems ◽  
Np Complete ◽  
Feasible Solutions ◽  
Np Problems

The theory of computational complexity is concerned with estimating the resources a computer needs to solve a given problem. The basic resources are time (number of steps executed) and space (amount of memory used). There are problems in logic, algebra and combinatorial games that are solvable in principle by a computer, but computationally intractable because the resources required by relatively small instances are practically infeasible. The theory of NP-completeness concerns a common type of problem in which a solution is easy to check but may be hard to find. Such problems belong to the class NP; the hardest ones of this type are the NP-complete problems. The problem of determining whether a formula of propositional logic is satisfiable or not is NP-complete. The class of problems with feasible solutions is commonly identified with the class P of problems solvable in polynomial time. Assuming this identification, the conjecture that some NP problems require infeasibly long times for their solution is equivalent to the conjecture that P≠NP. Although the conjecture remains open, it is widely believed that NP-complete problems are computationally intractable.

scholarly journals Total Weight Choosability of Graphs

10.37236/599 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jakub Przybyło ◽  
Mariusz Woźniak
Keyword(s):  
Main Tool ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Total Weight ◽  
Combinatorial Nullstellensatz ◽  
Complete Graphs ◽  
Complete Bipartite Graphs ◽  
Proper Vertex ◽  
Vertex Colouring

Suppose the edges and the vertices of a simple graph $G$ are assigned $k$-element lists of real weights. By choosing a representative of each list, we specify a vertex colouring, where for each vertex its colour is defined as the sum of the weights of its incident edges and the weight of the vertex itself. How long lists ensures a choice implying a proper vertex colouring for any graph? Is there any finite bound or maybe already lists of length two are sufficient? We prove that $2$-element lists are enough for trees, wheels, unicyclic and complete graphs, while the ones of length $3$ are sufficient for complete bipartite graphs. Our main tool is an algebraic theorem by Alon called Combinatorial Nullstellensatz.

scholarly journals The Harmonious Chromatic Number of Almost All Trees

1995 ◽  
Vol 4 (1) ◽  
pp. 31-46 ◽  
Author(s):  
Keith Edwards
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Simple Graph ◽  
Almost All ◽  
Proper Vertex ◽  
Vertex Colouring

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring.For any positive integer m, let Q(m) be the least positive integer k such that ≥ m. We show that for almost all unlabelled, unrooted trees T, h(T) = Q(m), where m is the number of edges of T.

TERM EQUATION SATISFIABILITY OVER FINITE ALGEBRAS

2010 ◽  
Vol 20 (08) ◽  
pp. 1001-1020 ◽  
Author(s):  
TOMASZ A. GORAZD ◽  
JACEK KRZACZKOWSKI
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Finite Algebra ◽  
Satisfiability Problem ◽  
Finite Algebras ◽  
Maximal Clones ◽  
Polynomial Time Algorithms ◽  
Np Complete

We study the computational complexity of the satisfiability problem of an equation between terms over a finite algebra (TERM-SAT). We describe many classes of algebras where the complexity of TERM-SAT is determined by the clone of term operations. We classify the complexity for algebras generating maximal clones. Using this classification we describe many of algebras where TERM-SAT is NP-complete. We classify the situation for clones which are generated by an order or a permutation relation. We introduce the concept of semiaffine algebras and show polynomial-time algorithms which solve the satisfiability problem for them.

scholarly journals Weighted Electoral Control

2015 ◽  
Vol 52 ◽  
pp. 507-542 ◽  
Author(s):  
Piotr Faliszewski ◽  
Edith Hemaspaandra ◽  
Lane A. Hemaspaandra
Keyword(s):  
Approximation Algorithms ◽  
Polynomial Time ◽  
Exact Algorithms ◽  
Weighted Voting ◽  
Np Completeness ◽  
Polynomial Time Algorithms ◽  
Np Complete ◽  
Electoral Control ◽  
Natural Settings ◽  
Work Done

Although manipulation and bribery have been extensively studied under weighted voting, there has been almost no work done on election control under weighted voting. This is unfortunate, since weighted voting appears in many important natural settings. In this paper, we study the complexity of controlling the outcome of weighted elections through adding and deleting voters. We obtain polynomial-time algorithms, NP-completeness results, and for many NP-complete cases, approximation algorithms. In particular, for scoring rules we completely characterize the complexity of weighted voter control. Our work shows that for quite a few important cases, either polynomial-time exact algorithms or polynomial-time approximation algorithms exist.

scholarly journals On Harmonious Colouring of Trees

10.37236/9 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
A. Aflaki ◽  
S. Akbari ◽  
K.J. Edwards ◽  
D.S. Eskandani ◽  
M. Jamaali ◽  
...  
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Simple Graph ◽  
Delta G ◽  
Proper Vertex ◽  
Vertex Colouring

Let $G$ be a simple graph and $\Delta(G)$ denote the maximum degree of $G$. A harmonious colouring of $G$ is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number $h(G)$ is the least number of colours in such a colouring. In this paper it is shown that if $T$ is a tree of order $n$ and $\Delta(T)\geq\frac{n}{2}$, then there exists a harmonious colouring of $T$ with $\Delta(T)+1$ colours such that every colour is used at most twice. Thus $h(T)=\Delta(T)+1$. Moreover, we prove that if $T$ is a tree of order $n$ and $\Delta(T) \le \Big\lceil\frac{n}{2}\Big\rceil$, then there exists a harmonious colouring of $T$ with $\Big\lceil \frac{n}{2}\Big \rceil +1$ colours such that every colour is used at most twice. Thus $h(T)\leq \Big\lceil \frac{n}{2} \Big\rceil +1$.

Sign in / Sign up

Export Citation Format

Share Document