scholarly journals Coloring Problems on Bipartite Graphs of Small Diameter

10.37236/9931 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Victor A. Campos ◽  
Guilherme C.M. Gomes ◽  
Allen Ibiapina ◽  
Raul Lopes ◽  
Ignasi Sau ◽  
...  
Keyword(s):  
Small Diameter ◽  
Bipartite Graphs ◽  
List Coloring ◽  
Coloring Problem ◽  
Np Completeness ◽  
Np Complete

We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we investigate the $k$-List Coloring, $k$-Coloring, and $k$-Precoloring Extension problems on bipartite graphs with diameter at most $d$, proving $\textsf{NP}$-completeness in most cases, and leaving open only the List $3$-Coloring and $3$-Precoloring Extension problems when $d=3$. Some of these results are obtained $\textsc{through}$ a proof that the Surjective $C_6$-Homomorphism problem is $\textsf{NP}$-complete on bipartite graphs with diameter at most four. Although the latter result has been already proved [Vikas, 2017], we present ours as an alternative simpler one. As a byproduct, we also get that $3$-Biclique Partition is $\textsf{NP}$-complete. An attempt to prove this result was presented in [Fleischner, Mujuni, Paulusma, and Szeider, 2009], but there was a flaw in their proof, which we identify and discuss here. Finally, we prove that the $3$-Fall Coloring problem is $\textsf{NP}$-complete on bipartite graphs with diameter at most four, and prove that $\textsf{NP}$-completeness for diameter three would also imply $\textsf{NP}$-completeness of $3$-Precoloring Extension on diameter three, thus closing the previously mentioned open cases. This would also answer a question posed in [Kratochvíl, Tuza, and Voigt, 2002].

Vertex-Neighbor-Scattering Number of Bipartite Graphs

2016 ◽  
Vol 27 (04) ◽  
pp. 501-509
Author(s):  
Zongtian Wei ◽  
Nannan Qi ◽  
Xiaokui Yue
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Bipartite Graphs ◽  
Upper And Lower Bounds ◽  
Number Of Components ◽  
Np Completeness ◽  
Vertex Set ◽  
Np Complete

Let G be a connected graph. A set of vertices [Formula: see text] is called subverted from G if each of the vertices in S and the neighbor of S in G are deleted from G. By G/S we denote the survival subgraph that remains after S is subverted from G. A vertex set S is called a cut-strategy of G if G/S is disconnected, a clique, or ø. The vertex-neighbor-scattering number of G is defined by [Formula: see text], where S is any cut-strategy of G, and ø(G/S) is the number of components of G/S. It is known that this parameter can be used to measure the vulnerability of spy networks and the computing problem of the parameter is NP-complete. In this paper, we discuss the vertex-neighbor-scattering number of bipartite graphs. The NP-completeness of the computing problem of this parameter is proven, and some upper and lower bounds of the parameter are also given.

Reconstructing a Graph from its Neighborhood Lists

1993 ◽  
Vol 2 (2) ◽  
pp. 103-113 ◽  
Author(s):  
Martin Aigner ◽  
Eberhard Triesch
Keyword(s):  
Decision Problem ◽  
Graph Isomorphism ◽  
Bipartite Graphs ◽  
Labeled Graph ◽  
Np Complete

Associate to a finite labeled graph G(V, E) its multiset of neighborhoods (G) = {N(υ): υ ∈ V}. We discuss the question of when a list is realizable by a graph, and to what extent G is determined by (G). The main results are: the decision problem is NP-complete; for bipartite graphs the decision problem is polynomially equivalent to Graph Isomorphism; forests G are determined up to isomorphism by (G); and if G is connected bipartite and (H) = (G), then H is completely described.

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.

scholarly journals FOLDING EQUILATERAL PLANE GRAPHS

2013 ◽  
Vol 23 (02) ◽  
pp. 75-92 ◽  
Author(s):  
ZACHARY ABEL ◽  
ERIK D. DEMAINE ◽  
MARTIN L. DEMAINE ◽  
SARAH EISENSTAT ◽  
JAYSON LYNCH ◽  
...  
Keyword(s):  
Plane Graph ◽  
Analogous Problem ◽  
Central Vertex ◽  
Polyhedral Complex ◽  
Plane Graphs ◽  
Single Vertex ◽  
Np Completeness ◽  
Np Complete ◽  
Polyhedral Manifold

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

scholarly journals Dynamic list coloring of bipartite graphs

2010 ◽  
Vol 158 (17) ◽  
pp. 1963-1965 ◽  
Author(s):  
Louis Esperet
Keyword(s):  
Bipartite Graphs ◽  
List Coloring

COMPLEXITY AND APPROXIMABILITY OF DOUBLE DIGEST

2005 ◽  
Vol 03 (02) ◽  
pp. 207-223
Author(s):  
MARK CIELIEBAK ◽  
STEPHAN EIDENBENZ ◽  
GERHARD J. WOEGINGER
Keyword(s):  
Real Life ◽  
Approximation Ratio ◽  
Additional Restriction ◽  
Np Hard ◽  
Model Errors ◽  
Np Completeness ◽  
The Third ◽  
Np Complete ◽  
Feasible Solutions

We revisit the DOUBLE DIGEST problem, which occurs in sequencing of large DNA strings and consists of reconstructing the relative positions of cut sites from two different enzymes. We first show that DOUBLE DIGEST is strongly NP-complete, improving upon previous results that only showed weak NP-completeness. Even the (experimentally more meaningful) variation in which we disallow coincident cut sites turns out to be strongly NP-complete. In the second part, we model errors in data as they occur in real-life experiments: we propose several optimization variations of DOUBLE DIGEST that model partial cleavage errors. We then show that most of these variations are hard to approximate. In the third part, we investigate variations with the additional restriction that coincident cut sites are disallowed, and we show that it is NP-hard to even find feasible solutions in this case, thus making it impossible to guarantee any approximation ratio at all.

scholarly journals On a list-coloring problem

2003 ◽  
Vol 268 (1-3) ◽  
pp. 303-308 ◽  
Author(s):  
Sylvain Gravier ◽  
Frédéric Maffray ◽  
Bojan Mohar
Keyword(s):  
List Coloring ◽  
Coloring Problem

On the Hardness of Devising Interval Routing Schemes

1997 ◽  
Vol 07 (01) ◽  
pp. 39-47 ◽  
Author(s):  
Michele Flammini
Keyword(s):  
Time Complexity ◽  
Compact Routing ◽  
Routing Schemes ◽  
Np Completeness ◽  
Routing Scheme ◽  
Np Complete ◽  
Interval Routing ◽  
General Networks

The k-Interval Routing Scheme (k-IRS) is a compact routing scheme on general networks. It has been studied extensively and recently been implemented on the latest generation of the INMOS transputer router chips. In this paper we investigate the time complexity of devising a minimal space k-IRS and we prove that the problem of deciding whether there exists a 2-IRS for any network G is NP-complete. This is the first hardness result for k-IRS where k is constant and the graph underlying the network is unweighted. Moreover, the NP-completeness holds also for linear and strict 2-IRS.

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