In a finite undirected graph G = (V, E), a vertex v is an element of V dominates itself and its neighbors in G. A vertex set D subset of V is an efficient dominating set (e.d.s. for short) of G if every v 2 V is domin...
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In a finite undirected graph G = (V, E), a vertex v is an element of V dominates itself and its neighbors in G. A vertex set D subset of V is an efficient dominating set (e.d.s. for short) of G if every v 2 V is dominated in G by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is known to be NP-complete for P-7-free graphs and solvable in polynomialtime for P-5-free graphs. The P-6-free case was the last open question for the complexity of ED on F-free graphs. Recently, Lokshtanov, Pilipczuk, and van Leeuwen showed that weighted ED is solvable in polynomialtime for P-6-free graphs, based on their quasi-polynomialalgorithm for the Maximum Weight Independent Set problem for P-6-free graphs. Independently, by a direct approach which is simpler and faster, we found an O(n(5)m) time solution for weighted ED on P-6-free graphs. Moreover, we show that weighted ED is solvable in linear time for P-5-free graphs which solves another open question for the complexity of (weighted) ED. The result for P-5-free graphs is based on modular decomposition.
Let k be a field. We are interested in the families of r-dimensional subspaces of k(n) with the following transversality property: any linear subspace of k(n) of dimension n - r is transversal to at least one element ...
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Let k be a field. We are interested in the families of r-dimensional subspaces of k(n) with the following transversality property: any linear subspace of k(n) of dimension n - r is transversal to at least one element of the family. While it is known how to build such families in polynomialtime over infinite fields k, no such technique is known for finite fields. However, transversal families in dimension n can be built when the field k is large enough with respect to n. We improve here on how large k needs to be with respect to the considered dimension n. (C) 2008 Elsevier Inc. All rights reserved.
A subcoloring is a vertex coloring of a graph in which every color class induces a disjoint union of cliques. We derive a number of results on the combinatorics, the algorithmics, and the complexity of subcolorings. O...
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A subcoloring is a vertex coloring of a graph in which every color class induces a disjoint union of cliques. We derive a number of results on the combinatorics, the algorithmics, and the complexity of subcolorings. On the negative side, we prove that 2-subcoloring is NP-hard for comparability graphs, and that 3-subcoloring is NP-hard for AT-free graphs and for complements of planar graphs. On the positive side, we derive polynomial time algorithms for 2-subcoloring of complements of planar graphs, and for r-subcoloring of interval and of permutation graphs. Moreover, we prove asymptotically best possible upper bounds on the subchromatic number-of interval graphs, chordal graphs, and permutation graphs in terms of the number of vertices.
In this article. we study a class of new scheduling models where time slot costs have to be taken into consideration. In Such models, processing a job will incur certain cost which is determined by the time slots occu...
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In this article. we study a class of new scheduling models where time slot costs have to be taken into consideration. In Such models, processing a job will incur certain cost which is determined by the time slots occupied by the job in a schedule. The models apply when operational costs vary over time. The objective of the scheduling models is to minimize the total time slot costs plus a traditional scheduling performance measure. We consider the following performance measures: total completion time, maximum lateness/tardiness, total weighted number of tardy jobs, and total tardiness. We prove the intractability of the models under general parameters and provide polynomial-timealgorithms for special cases with non-increasing time slot costs. (C) 2010 Wiley Periodicals. Inc. Naval Research Logistics 57: 159-171, 2010
Let Gbe a vertex-colored connected graph. A subset X of the vertex-set of G is called proper if any two adjacent vertices in X have distinct colors. The graph G is called proper vertex-disconnected if for any two vert...
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Let Gbe a vertex-colored connected graph. A subset X of the vertex-set of G is called proper if any two adjacent vertices in X have distinct colors. The graph G is called proper vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, Sis proper and x and y belong to different components of G - S;where as when x and y are adjacent, S + x or S + y is proper and x and y belong to different components of (G - xy) - S. For a connected graph G, the proper vertex-disconnection number of G, denoted by pvd( G), is the minimum number of colors that are needed to make G proper vertex-disconnected. In this paper, we firstly characterize the graphs of order n with proper vertex-disconnection number k for k.{1, n - 2, n - 1, n}. Secondly, we give some sufficient conditions for a graph G such that pvd(G) =.(G), and show that almost all graphs G have pvd(G) =.(G) and pvd(G) =.(G). We also give an equivalent statement of the famous Four Color Theorem. Furthermore, we study the relationship between the proper disconnection number pd(G) of G and the proper vertex-disconnection number pvd(L(G)) of the line graph L(G) of G. Finally, we show that it is NP-complete to decide whether a given vertex-colored graph is proper vertex-disconnected, and it is NP-hard to decide for a fixed integer k = 3, whether the pvd-number of a graph G is no more than k, even if k = 3 and G is a planar graph with Delta(G) = 12. We also show that it is solvable in polynomialtime to determine the proper vertex-disconnection number for a graph with maximum degree less than four. (c) 2022 Elsevier B.V. All rights reserved.
A perfect Roman dominating function on a graph G is a function f : V (G) -> {0,1, 2} having the property that for every vertex u with f (u) = 0, there exists exactly one vertex v such that uv is an element of E(G) ...
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A perfect Roman dominating function on a graph G is a function f : V (G) -> {0,1, 2} having the property that for every vertex u with f (u) = 0, there exists exactly one vertex v such that uv is an element of E(G) and f (v) = 2. The weight of f, denoted by w(f), is the value Sigma(v is an element of v(G)) f (v). Given a graph G and a positive integer k, the perfect Roman domination problem is to decide whether there is a perfect Roman dominating function f on G such that w(f) is at most k. In this paper, we first show that the perfect Roman domination problem is NP-complete for chordal graphs, planar graphs, and bipartite graphs. Then we present polynomial time algorithms for computing a perfect Roman dominating function with minimum weight in block graphs, cographs, series-parallel graphs, and proper interval graphs. (C) 2019 Elsevier B.V. All rights reserved.
Given an edge weighted graph, and an acyclic edge set, the target of the partial inverse maximum spanning tree problem (PIMST) is to get a new weight function such that the given set is included in some maximum spanni...
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Given an edge weighted graph, and an acyclic edge set, the target of the partial inverse maximum spanning tree problem (PIMST) is to get a new weight function such that the given set is included in some maximum spanning tree associated with the new function, and the difference between the two functions is minimum. In this paper, we research PIMST under the Chebyshev norm. Firstly, the definition of extreme optimal solution is introduced, and its some properties are gained. Based on these properties, a polynomial scale optimal value candidate set is obtained. Finally, strongly polynomial-timealgorithms for solving this problem are proposed. Thus, the computational complexity of PIMST is completely solved.
We investigate a curious problem from additive number theory: Given two positive integers S and Q, does there exist a sequence of positive integers that add up to S and whose squares add up to Q? We show that this pro...
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We investigate a curious problem from additive number theory: Given two positive integers S and Q, does there exist a sequence of positive integers that add up to S and whose squares add up to Q? We show that this problem can be solved in timepolynomially bounded in the logarithms of S and Q. As a consequence, also the following question can be answered in polynomialtime: For given numbers n and m, do there exist n lines in the Euclidean plane with exactly m points of intersection? (C) 2004 Published by Elsevier B.V.
Let G = (V, E) be a finite undirected graph without loops and multiple edges. A subset M subset of E of edges is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of M. ...
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Let G = (V, E) be a finite undirected graph without loops and multiple edges. A subset M subset of E of edges is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of M. In particular, this means that M is an induced matching, and every edge not in M shares exactly one vertex with an edge in M. Clearly, not every graph has a d.i.m. The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G;this problem is also known as the Efficient Edge Domination problem;it is the Efficient Domination problem for line graphs. The DIM problem is NP-complete in general, and even for very restricted graph classes such as planar bipartite graphs with maximum degree 3. However, DIM is solvable in polynomialtime for claw-free (i.e., S-1,S-1,S-1-free) graphs, for S-1,S-2,S-3-free graphs, for S-2,S-2,S-2-free graphs as well as for S-2,S-2,S-3-free graphs, in linear time for P-7-free graphs, and in polynomialtime for P-8-free graphs (P-k is a special case of S-i,S-j,S-l). In a paper by Hertz, Lozin, Ries, Zamaraev and de Werra, it was conjectured that DIM is solvable in polynomialtime for S-i,S-j,S-k-free graphs for every fixed i, j, k. In this paper, combining two distinct approaches, we solve it in polynomialtime for S-1,S-2,S-4-free graphs which generalizes the S-1,S-2,S-3-free as well as the P7-free case. (C) 2018 Elsevier B.V. All rights reserved.
This paper studies preemptive bi-criteria scheduling on m parallel machines with machine unavailable intervals. The goal is to minimize the total completion time subject to the constraint that the makespan is at most ...
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This paper studies preemptive bi-criteria scheduling on m parallel machines with machine unavailable intervals. The goal is to minimize the total completion time subject to the constraint that the makespan is at most a constant T. We study the unavailability model such that the number of available machines cannot go down by 2 within any period of p(max) where p(max) is the maximum processing time among all jobs. We show that there is an optimal polynomial time algorithm. (C) 2015 Elsevier B.V. All rights reserved.
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