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## The complexity of the edge 3-colorability problem for graphs without two induced fragments each on at most six vertices

Siberian Electronic Mathematical Reports. 2014. Vol. 11. P. 811-822.

We obtain a complete complexity dichotomy for the edge 3- colorability within the family of hereditary classes defined by forbidden

induced subgraphs on at most 6 vertices and having at most two 6-vertex forbidden induced structures.

Language:
English

Publication based on the results of:

Malyshev D., Discrete Mathematics and Applications 2017 Vol. 27 No. 2 P. 97-101

A class of graphs is called monotone if it is closed under deletion of vertices and edges. Any such class may be defined in terms of forbidden subgraphs. The chromatic index of a graph is the smallest number of colors required for its edge-coloring such that any two adjacent edges have different colors. We obtain ...

Added: May 10, 2017

Malyshev D., Pardalos P. M., Optimization Letters 2016 Vol. 10 No. 8 P. 1593-1612

The task of complete complexity dichotomy is to clearly distinguish between easy and hard cases of a given problem on a family of subproblems. We consider this task for some optimization problems restricted to certain classes of graphs closed under deletion of vertices. A concept in the solution process is based on revealing the so-called ...

Added: December 18, 2015

Malyshev D., Discrete Applied Mathematics 2016 Vol. 203 P. 117-126

We completely determine the complexity status of the dominating set problem for hereditary graph classes defined by forbidden induced subgraphs with at most five vertices. ...

Added: October 9, 2015

Goldengorin B. I., Malyshev D., Pardalos P. M., Doklady Mathematics 2013 Vol. 87 No. 3 P. 368-371

The notion of a tolerance of an element of a combinatorial optimization problem is often used for stability analysis of an optimal solution and it is a base for design branch-and-bound algorithms solving such problems. In this paper we show that for the weighted independent set problem on trees with n vertices all upper and ...

Added: June 23, 2013

Malyshev D., Razvenskaya O., Discrete Applied Mathematics 2017 Vol. 219 P. 158-166

We show that the chromatic number of {P_5,K_p-e}-free graphs can be computed in polynomial time for each fixed p.
Additionally, we prove polynomial-time solvability of the weighted vertex coloring problem for {P_5,co(P_3+P_2)}-free graphs. ...

Added: November 21, 2016

Malyshev D., Pardalos P. M., Doklady Mathematics 2014 Vol. 89 No. 2 P. 253-256

The notion of a tolerance of an element of a combinatorial optimization problem is often used for stability analysis of an
optimal solution and it is a base for design of branch-and-bound algorithms solving such problems. We show in this paper that for the weighted independent set problem and a bipartite graph with n vertices and ...

Added: April 18, 2014

Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2013 Vol. 7 No. 4 P. 537-548

We prove the polynomial solvability of the independent set problem for some family of
classes of the planar subcubic graphs. ...

Added: January 21, 2014

Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2014 Vol. 8 No. 2 P. 245-255

The edge list-ranking problem is a generalization of the classical edge coloring problem, and it is a mathematical model for some parallel processes. The computational complexity of this problem is under study for graph sets closed under isomorphism and deletion of vertices (hereditary classes). Allfinitely defined and minor-closed cases are described for which the problem ...

Added: May 8, 2014

Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2013 Vol. 7 No. 3 P. 412-419

The notion is introduced of an expanding operator for the independent set problem. This notion is a useful tool for the constructive formation of new cases with the efficient solvability of the problem in the family of hereditary classes of graphs and is applied to hereditary parts of the set Free({P_5,C_5}). It is proved that ...

Added: October 3, 2013

Malyshev D., Discrete Applied Mathematics 2018 Vol. 247 P. 423-432

We show that the weighted coloring problem can be solved for {P5,banner}-free graphs and for {P5,dart}-free graphs in polynomial time on the sum of vertex weights. ...

Added: April 23, 2018

Malyshev D., Optimization Letters 2021 Vol. 15 No. 2 P. 311-326

The vertex colourability problem is to determine, for a given graph and a given natural k, whether it is possible to split the graph’s vertex set into at most k subsets, each of pairwise non-adjacent vertices, or not. A hereditary class is a set of simple graphs, closed under deletion of vertices. Any such a class can be ...

Added: January 6, 2021

Malyshev D., / Cornell University. Series math "arxiv.org". 2013. No. 1307.0278v1.

The coloring problem is studied in the paper for graph classes deﬁned by two small forbidden induced subgraphs. We prove some suﬃcient conditions for eﬀective solvability of the problem in such classes. As their corollary we determine the computational complexity for all sets of two connected forbidden induced subgraphs with at most ﬁve vertices except ...

Added: October 3, 2013

Malyshev D., Gribanov D., Discrete Optimization 2018 Vol. 29 P. 103-110

We consider boolean linear programming formulations of the vertex and edge dominating set problems and prove their polynomial-time solvability for classes of graphs with constraint matrices having bounded minors in the absolute value. ...

Added: April 8, 2018

Malyshev D., Дискретный анализ и исследование операций 2012 Т. 19 № 3 С. 58-64

An algorithm is implemented in the article for finding the independence number of a n-vertex graph from the class Free({P5,C5, Kp}) in time O(np+O(1)). ...

Added: June 6, 2012

Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2020 Vol. 14 No. 4 P. 706-721

The edge coloring problem for a graph is to minimize the number of colors that are sufficient to color all edges of the graph so that all adjacent edges receive distinct colors. The computational complexity of the problem is known for all graph classes defined by forbidden subgraphs with at most 6 edges. We improve ...

Added: January 30, 2021

Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2017 Vol. 11 No. 1 P. 99-106

The notions of boundary and minimal hard classes of graphs, united by the term “critical classes”, are useful tools for analysis of computational complexity of graph problems in the family of hereditary graph classes. In this family, boundary classes are known for several graph problems. In the paper, we consider critical graph classes in the ...

Added: February 13, 2017

Malyshev D., Pardalos P. M., Optimization Letters 2015 Vol. 9 No. 5 P. 839-843

The quadratic programming problem is known to be NP-hard for Hessian matrices with only one negative eigenvalue, but it is tractable for convex instances. These facts yield to consider the number of negative eigenvalues as a complexity measure
of quadratic programs. We prove here that the clique problem is tractable for two variants of its Motzkin-Strauss ...

Added: September 26, 2014

Shvydun S., / Высшая школа экономики. Series WP7 "Математические методы анализа решений в экономике, бизнесе и политике". 2015. No. WP7/2015/07.

Two-stage superposition choice procedures, which sequentially apply two choice procedures so that the result of the first choice procedure is the input for the second choice procedure, are studied. We define which of them satisfy given normative conditions, showing how a final choice is changed due to the changes of preferences or a set of ...

Added: October 20, 2015

Gribanov D., Malyshev D., Discrete Applied Mathematics 2017 Vol. 227 P. 13-20

We consider boolean linear programming formulations of the independent set, the vertex and the edge dominating set problems and prove their polynomial-time solvability for classes of graphs with (augmented) constraint matrices having bounded minors in the absolute value ...

Added: April 23, 2017

Malyshev D., Discrete Mathematics 2015 Vol. 338 No. 11 P. 1860-1865

We completely determine the complexity status of the 3-colorability problem for hereditary graph classes defined by two forbidden induced subgraphs with at most five vertices. ...

Added: April 7, 2014

Kontchakov R., Pratt-Hartmann I., Nenov Y. et al., ACM Transactions on Computational Logic 2013 Vol. 14 No. 2 P. 13.1-13.48

We consider the quantifier-free languages, Bc and Bc°, obtained by augmenting the signature of Boolean algebras with a unary predicate representing, respectively, the property of being connected, and the property of having a connected interior. These languages are interpreted over the regular closed sets of Rn (n ≥ 2) and, additionally, over the regular closed ...

Added: March 25, 2015

Kazda A., Opršal J., Valeriote M. et al., Canadian Mathematical Bulletin 2020 P. 577-591

This paper investigates the computational complexity of deciding if a given finite idempotent algebra has a ternary term operation m that satisfies the minority equations m(y,x,x)≈m(x,y,x)≈m(x,x,y)≈y . We show that a common polynomial-time approach to testing for this type of condition will not work in this case and that this decision problem lies in the class NP. ...

Added: June 15, 2020

Korpelainen N., Lozin V. V., Malyshev D. et al., Theoretical Computer Science 2011 No. 412 P. 3545-3554

The notion of a boundary graph property was recently introduced as a relaxation of that of a minimal property and was applied to several problems of both algorithmic and combinatorial nature. In the present paper, we first survey recent results related to this notion and then apply it to two algorithmic graph problems: Hamiltonian cycle ...

Added: September 11, 2012

Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2012 Vol. 6 No. 1 P. 97-99

Under study is the complexity status of the independent set problem in a class of connected graphs that are defined by functional constraints on the number of edges depending on the number of vertices. For every natural number C, this problem is shown to be polynomially solvable in the class of graphs, On the other ...

Added: December 7, 2012