38 research outputs found
Independent domination in hereditary classes
AbstractWe investigate Independent Domination Problem within hereditary classes of graphs. Boliac and Lozin [Independent domination in finitely defined classes of graphs, Theoret. Comput. Sci. 301 (1–3) (2003) 271–284] proved some sufficient conditions for Independent Domination Problem to be NP-complete within finitely defined hereditary classes of graphs. They posed a question whether the conditions are also necessary. We show that the conditions are not necessary, since Independent Domination Problem is NP-hard within 2P3-free graphs.Moreover, we show that the problem remains NP-hard for a new hereditary class of graphs, called hereditary 3-satgraphs. We characterize hereditary 3-satgraphs in terms of forbidden induced subgraph. As corollaries, we prove that Independent Domination Problem is NP-hard within the class of all 2P3-free perfect graphs and for K1,5-free weakly chordal graphs.Finally, we compare complexity of Independent Domination Problem with that of Independent Set Problem for a hierarchy of hereditary classes recently proposed by Hammer and Zverovich [Construction of maximal stable sets with k-extensions, Combin. Probab. Comput. 13 (2004) 1–8]. For each class in the hierarchy, a maximum independent set can be found in polynomial time, and the hierarchy covers all graphs. However, our characterization of hereditary 3-satgraphs implies that Independent Domination Problem is NP-hard for almost all classes in the hierarchy. This fact supports a conjecture that Independent Domination is harder than Independent Set Problem within hereditary classes
List coloring in the absence of a linear forest.
The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The Listk-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u)⊆{1,…,k}. Let Pn denote the path on n vertices, and G+H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that Listk-Coloring can be solved in polynomial time for graphs with no induced rP1+P5, hereby extending the result of Hoà ng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1+P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H
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Some Problems in Graph Theory and Scheduling
In this dissertation, we present three results related to combinatorial algorithms in graph theory and scheduling, both of which are important subjects in the area of discrete mathematics and theoretical computer science. In graph theory, a graph is a set of vertices and edges, where each edge is a pair of vertices. A coloring of a graph is a function that assigns each vertex a color such that no two adjacent vertices share the same color. The first two results are related to coloring graphs belonging to specific classes. In scheduling problems, we are interested in how to efficiently schedule a set of jobs on machines. The last result is related to a scheduling problem in an environment where there is uncertainty on the number of machines.
The first result of this thesis is a polynomial time algorithm that determines if an input graph containing no induced seven-vertex path is 3-colorable. This affirmatively answers a question posed by Randerath, Schiermeyer and Tewes in 2002. Our algorithm also solves the list-coloring version of the 3-coloring problem, where every vertex is assigned a list of colors that is a subset of {1, 2, 3}, and gives an explicit coloring if one exists. This is joint work with Flavia Bonomo, Maria Chundnovsky, Peter Maceli, Oliver Schaudt, and Maya Stein.
A graph is H-free if it has no induced subgraph isomorphic to H. In the second part of this thesis, we characterize all graphs for which there are only finitely many minimal non-three-colorable H-free graphs. This solves a problem posed by Golovach et al. We also characterize all graphs H for which there are only finitely many H-free minimal obstructions for list 3-colorability. This is joint work with Maria Chudnovsky, Jan Goedgebeur and Oliver Schaudt.
The last result of this thesis deals with a scheduling problem addressing the uncertainty regarding the machines. We study a scheduling environment in which jobs first need to be grouped into some sets before the number of machines is known, and then the sets need to be scheduled on machines without being separated. In order to evaluate algorithms in such an environment, we introduce the idea of an alpha-robust algorithm, one which is guaranteed to return a schedule on any number m of machines that is within an alpha factor of the optimal schedule on m machines, where the optimum is not subject to the restriction that the sets cannot be separated. Under such environment, we give a (5/3+epsilon)-robust algorithm for scheduling on parallel machines to minimize makespan, and show a lower bound of 4/3. For the special case when the jobs are infinitesimal, we give a 1.233-robust algorithm with an asymptotic lower bound of 1.207. This is joint work with Clifford Stein
Odd Wheels Are Not Odd-Distance Graphs
An odd wheel graph is a graph formed by connecting a new vertex to all vertices of an odd cycle. We answer a question of Rosenfeld and Le by showing that odd wheels cannot be drawn in the plane so that the lengths of the edges are odd integers