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Search results for: COMPLETE BIPARTITE GRAPHS
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On incidence coloring of coloring of complete multipartite and semicubic bipartite graphs
PublicationIn the paper, we show that the incidence chromatic number of a complete k-partite graph is at most ∆+2 (i.e., proving the incidence coloring conjecture for these graphs) and it is equal to ∆+1 if and only if the smallest part has only one vertex.
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On the deficiency of bipartite graphs
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Interval incidence coloring of bipartite graphs
PublicationIn this paper we study the problem of interval incidence coloring of bipartite graphs. We show the upper bound for interval incidence coloring number (χii) for bipartite graphs χii≤2Δ, and we prove that χii=2Δ holds for regular bipartite graphs. We solve this problem for subcubic bipartite graphs, i.e. we fully characterize the subcubic graphs that admit 4, 5 or 6 coloring, and we construct a linear time exact algorithm for subcubic...
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Chromatic cost coloring of weighted bipartite graphs
PublicationGiven a graph G and a sequence of color costs C, the Cost Coloring optimization problem consists in finding a coloring of G with the smallest total cost with respect to C. We present an analysis of this problem with respect to weighted bipartite graphs. We specify for which finite sequences of color costs the problem is NP-hard and we present an exact polynomial algorithm for the other finite sequences. These results are then extended...
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Bipartite theory of graphs: outer-independent domination
PublicationLet $G = (V,E)$ be a bipartite graph with partite sets $X$ and $Y$. Two vertices of $X$ are $X$-adjacent if they have a common neighbor in $Y$, and they are $X$-independent otherwise. A subset $D \subseteq X$ is an $X$-outer-independent dominating set of $G$ if every vertex of $X \setminus D$ has an $X$-neighbor in $D$, and all vertices of $X \setminus D$ are pairwise $X$-independent. The $X$-outer-independent domination number...
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Sum Coloring of Bipartite Graphs with Bounded Degree
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Sum coloring of bipartite graphs with bounded degree.
PublicationArtykuł poświęcony jest złożoności obliczeniowej zagadnienia sumacyjnego kolorowania grafów dwudzielnych o ograniczonym stopniu. Zawiera dowód tego, że sumacyjne kolorowanie grafów dwudzielnych stopnia mniejszego równego 5 jest NP-zupełne oraz opis wielomianowego algorytmu, który optymalnie sumacyjnie koloruje grafy dwudzielne podkubiczne.
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Scheduling on Uniform and Unrelated Machines with Bipartite Incompatibility Graphs
PublicationThe problem of scheduling jobs on parallel machines under an incompatibility relation is considered in this paper. In this model, a binary relation between jobs is given and no two jobs that are in the relation can be scheduled on the same machine. We consider job scheduling under the incompatibility relation modeled by a bipartite graph, under the makespan optimality criterion, on uniform and unrelated machines. Unrelated machines...
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A note on the strength and minimum color sum of bipartite graphs
PublicationSiłą grafu G nazywamy najmniejszą liczbę całkowitą s, taką że istniej pokolorowanie grafu G, o minimalnej sumie przy użyciu kolorów {1,...,s}. W pracy pokazano, że w grafach dwudzielnych stopnia D zachodzi oszacowanie s <= ceil(D/2) + 1. Z obserwacji tej wynika algorytm wielomianowy do obliczania siły i sumy chromatycznej w grafach dwudzielnych stopnia co najwyżej 4.
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Interval Edge Coloring of Bipartite Graphs with Small Vertex Degrees
PublicationAn edge coloring of a graph G is called interval edge coloring if for each v ∈ V(G) the set of colors on edges incident to v forms an interval of integers. A graph G is interval colorable if there is an interval coloring of G. For an interval colorable graph G, by the interval chromatic index of G, denoted by χ'_i(G), we mean the smallest number k such that G is interval colorable with k colors. A bipartite graph G is called (α,β)-biregular...
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The maximum edge-disjoint paths problem in complete graphs
PublicationRozważono problem ścieżek krawędziowo rozłącznych w grafach pełnych. Zaproponowano wielomianowe algorytmy: 3.75-przybliżony (off-line) oraz 6.47-przybliżony (on-line), poprawiając tym samym wyniki wcześniej znane z literatury [P. Carmi, T. Erlebach, Y. Okamoto, Greedy edge-disjoint paths in complete graphs, in: Proc. 29th Workshop on Graph Theoretic Concepts in Computer Science, in: LNCS, vol. 2880, 2003, pp. 143-155]. Ponadto...
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Ramsey numbers for triangles versus almost-complete graphs.
PublicationPokazano, że w każdym krawędziowym pokolorowaniu dwoma kolorami grafu pełnego o 38 wierzchołkach występuje trójkąt w pierwszym kolorze lub podgraf izomorficzny z K_10 - e w drugim kolorze. Stąd otrzymujemy górne oszacowanie R(K_3, K_10 - e) <= 38. Przedstawiamy także pokolorowanie krawędziowe grafu K_36, którego istnienie dowodzi, że R(K_3, K_10 - e) >= 37.
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A note on polynomial algorithm for cost coloring of bipartite graphs with Δ ≤ 4
PublicationIn the note we consider vertex coloring of a graph in which each color has an associated cost which is incurred each time the color is assigned to a vertex. The cost of coloring is the sum of costs incurred at each vertex. We show that the minimum cost coloring problem for n-vertex bipartite graph of degree ∆≤4 can be solved in O(n^2) time. This extends Jansen’s result [K.Jansen,The optimum cost chromatic partition problem, in:...
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Approximation strategies for routing edge disjoint paths in complete graphs
PublicationPraca dotyczy problemu ścieżek krawędziowo rozłącznych w nieskierowanych grafach pełnych, dla którego podano nowe algorytmy przybliżone: 3.75-przybliżony (model off-line) i 6.47-przybliżony (model on-line). Stosując podobną metodologię, uzyskano algorytm 4.5-przybliżony (off-line) i 6-przybliżony (on-line) dla problemu routingu i kolorowania ścieżek w grafach pełnych.
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Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines
PublicationThe problem of scheduling n identical jobs on 4 uniform machines with speeds s1>=s2>=s3>=s4 is considered.The aim is to find a schedule with minimum possible length. We assume that jobs are subject to mutual exclusion constraints modeled by a bipartite incompatibility graph of degree delta. We show that the general problem is NP-hard even if s1=s2=s3. If, however, delta<5 and s1>12s2 s2=s3=s4, then the problem can be solved to...
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Scheduling of identical jobs with bipartite incompatibility graphs on uniform machines. Computational experiments
PublicationWe consider the problem of scheduling unit-length jobs on three or four uniform parallel machines to minimize the schedule length or total completion time. We assume that the jobs are subject to some types of mutual exclusion constraints, modeled by a bipartite graph of a bounded degree. The edges of the graph correspond to the pairs of jobs that cannot be processed on the same machine. Although the problem is generally NP-hard,...
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The complexity of the L(p,q)-labeling problem for bipartite planar graphs of small degree
PublicationW pracy pokazano, że problem L(p,q)-kolorowania przy użyciu ''t'' kolorów jest NP-zupełny nawet w wersji ograniczonej do grafów planarnych dwudzielnych małego stopnia, nawet dla stosunkowo niewielkich wartości ''t''. Jako wniosek z uzyskanych wyników stwierdzono, że problem L(2,1)-kolorowania grafów planarnych przy użyciu 4 kolorów jest NP-zupełny, a także że problem L(p,q)-kolorowania grafów o maksymalnym stopniu 4 jest NP-zupełny...
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Better polynomial algorithms for scheduling unit-length jobs with bipartite incompatibility graphs on uniform machines
PublicationThe goal of this paper is to explore and to provide tools for the investigation of the problems of unit-length scheduling of incompatible jobs on uniform machines. We present two new algorithms that are a significant improvement over the known algorithms. The first one is Algorithm 2 which is 2-approximate for the problem Qm|p j = 1, G = bisubquartic|Cmax . The second one is Algorithm 3 which is 4-approximate for the problem Qm|p...
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Tight bounds on global edge and complete alliances in trees
PublicationIn the talk the authors present some tight upper bounds on global edge alliance number and global complete alliance number of trees. Moreover, we present our NP-completeness results from [8] for global edge alliances and global complete alliances on subcubic bipartite graphs without pendant vertices. We discuss also polynomial time exact algorithms for finding the minimum global edge alliance on trees [7] and complete alliance...
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Edge coloring of graphs of signed class 1 and 2
PublicationRecently, Behr (2020) introduced a notion of the chromatic index of signed graphs and proved that for every signed graph (G, σ) it holds that ∆(G) ≤ χ′(G,σ) ≤ ∆(G) + 1, where ∆(G) is the maximum degree of G and χ′ denotes its chromatic index. In general, the chromatic index of (G, σ) depends on both the underlying graph G and the signature σ. In the paper we study graphs G for which χ′(G, σ) does not depend on σ. To this aim we...
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A 27/26-approximation algorithm for the chromatic sum coloring of bipartitegraphs
PublicationWe consider the CHROMATIC SUM PROBLEM on bipartite graphs which appears to be much harder than the classical CHROMATIC NUMBER PROBLEM. We prove that the CHROMATIC SUM PROBLEM is NP-complete on planar bipartite graphs with Delta less than or equal to 5, but polynomial on bipartite graphs with Delta less than or equal to 3, for which we construct an O(n(2))-time algorithm. Hence, we tighten the borderline of intractability for this...
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The Backbone Coloring Problem for Bipartite Backbones
PublicationLet G be a simple graph, H be its spanning subgraph and λ≥2 be an integer. By a λ -backbone coloring of G with backbone H we mean any function c that assigns positive integers to vertices of G in such a way that |c(u)−c(v)|≥1 for each edge uv∈E(G) and |c(u)−c(v)|≥λ for each edge uv∈E(H) . The λ -backbone chromatic number BBCλ(G,H) is the smallest integer k such that there exists a λ -backbone coloring c of G with backbone H satisfying...
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Paired domination subdivision and multisubdivision numbers of graphs
PublicationThe paired domination subdivision number sdpr(G) of a graph G is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the paired domination number of G. We prove that the decision problem of the paired domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the paired domination muttisubdivision number of a nonempty graph...
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Domination subdivision and domination multisubdivision numbers of graphs
PublicationThe domination subdivision number sd(G) of a graph G is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of G. It has been shown [10] that sd(T)<=3 for any tree T. We prove that the decision problem of the domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the domination multisubdivision number...
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A construction for the hat problem on a directed graph
PublicationA team of n players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. Visibility is defined by a directed graph; that is, vertices correspond to players, and a player can see each player to whom he is connected by an arc. The...
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Some variations of perfect graphs
PublicationWe consider (ψk−γk−1)-perfect graphs, i.e., graphs G for which ψk(H) =γk−1(H) for any induced subgraph H of G, where ψk and γk−1 are the k -path vertex cover number and the distance (k−1)-domination number, respectively. We study (ψk−γk−1)-perfect paths, cycles and complete graphs for k≥2. Moreover, we provide a complete characterisation of (ψ2−γ1)-perfect graphs describing the set of its forbidden induced subgraphs and providing...
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Strategic balance in graphs
PublicationFor a given graph G, a nonempty subset S contained in V ( G ) is an alliance iff for each vertex v ∈ S there are at least as many vertices from the closed neighbourhood of v in S as in V ( G ) − S. An alliance is global if it is also a dominating set of G. The alliance partition number of G was defined in Hedetniemi et al. (2004) to be the maximum number of sets in a partition of V ( G ) such that each set is an alliance. Similarly,...
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Global edge alliances in graphs
PublicationIn the paper we introduce and study a new problem of finding a minimum global edge alliance in a graph which is related to the global defensive alliance (Haynes et al., 2013; Hedetniemi, 2004) and the global defensive set (Lewoń et al., 2016). We proved the NP-completeness of the global edge alliance problem for subcubic graphs and we constructed polynomial time algorithms for trees. We found the exact values of the size of the...
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The computational complexity of the backbone coloring problem for bounded-degree graphs with connected backbones
PublicationGiven a graph G, a spanning subgraph H of G and an integer λ>=2, a λ-backbone coloring of G with backbone H is a vertex coloring of G using colors 1, 2, ..., in which the color difference between vertices adjacent in H is greater than or equal to lambda. The backbone coloring problem is to find such a coloring with maximum color that does not exceed a given limit k. In this paper, we study the backbone coloring problem for bounded-degree...
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Minimum order of graphs with given coloring parameters
PublicationA complete k-coloring of a graph G=(V,E) is an assignment F: V -> {1,...,k} of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one edge. Three extensively investigated graph invariants related to complete colorings are the minimum and maximum number of colors in a complete coloring (chromatic number χ(G) and achromatic number ψ(G), respectively),...
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The hat problem on a union of disjoint graphs
PublicationThe topic is the hat problem in which each of n players is randomly fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of winning. In this version every player...
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The computational complexity of the backbone coloring problem for planar graphs with connected backbones
PublicationIn the paper we study the computational complexity of the backbone coloring problem for planar graphs with connected backbones. For every possible value of integer parameters λ≥2 and k≥1 we show that the following problem: Instance: A simple planar graph GG, its connected spanning subgraph (backbone) HH. Question: Is there a λ-backbone coloring c of G with backbone H such that maxc(V(G))≤k? is either NP-complete or polynomially...
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Equitable coloring of corona products of graphs
PublicationIn this paper we consider an equitable coloring of some corona products of graphs G and H in symbols, G o H). In particular, we show that deciding the colorability of G o H is NP-complete even if G is 4-regular and H is K_2. Next, we prove exact values or upper bounds on the equitable chromatic number of G o H, where G is an equitably 3- or 4-colorable graph and H is an r-partite graph, a path, a cycle or a complete graph.
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Algorithms for testing security in graphs
PublicationIn this paper we propose new algorithmic methods giving with the high probability the correct answer to the decision problem of security in graphs. For a given graph G and a subset S of a vertex set of G we have to decide whether S is secure, i.e. every subset X of S fulfils the condition: |N[X] \cap S| >= |N[X] \ S|, where N[X] is a closed neighbourhood of X in graph G. We constructed a polynomial time property pseudotester based...
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Progress on Roman and Weakly Connected Roman Graphs
PublicationA graph G for which γR(G)=2γ(G) is the Roman graph, and if γwcR(G)=2γwc(G), then G is the weakly connected Roman graph. In this paper, we show that the decision problem of whether a bipartite graph is Roman is a co-NP-hard problem. Next, we prove similar results for weakly connected Roman graphs. We also study Roman trees improving the result of M.A. Henning’s A characterization of Roman trees, Discuss. Math. Graph Theory 22 (2002)....
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Interval incidence graph coloring
PublicationIn this paper we introduce a concept of interval incidence coloring of graphs and survey its general properties including lower and upper bounds on the number of colors. Our main focus is to determine the exact value of the interval incidence coloring number χii for selected classes of graphs, i.e. paths, cycles, stars, wheels, fans, necklaces, complete graphs and complete k-partite graphs. We also study the complexity of the...
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Equitable coloring of corona multiproducts of graphs
PublicationWe give some results regarding the equitable chromatic number for l-corona product of two graphs: G and H, where G is an equitably 3- or 4-colorable graph and H is an r-partite graph, a cycle or a complete graph. Our proofs lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of G.
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Properties of the triset metric for phylogenetic trees
Publicationthe following paper presents a new polynomial time metric for unrootedphylogenetic trees (based on weighted bipartite graphs and the method ofdetermining a minimum perfect matching) and its properties. also many its properties are presented.
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Paired domination versus domination and packing number in graphs
PublicationGiven a graph G = (V(G), E(G)), the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph G are denoted by γ (G), γpr(G), and γt(G), respectively. For a positive integer k, a k-packing in G is a set S ⊆ V(G) such that for every pair of distinct vertices u and v in S, the distance between u and v is at least k + 1. The k-packing number is the order of a largest kpacking and...
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Interval incidence coloring of subcubic graphs
PublicationIn this paper we study the problem of interval incidence coloring of subcubic graphs. In [14] the authors proved that the interval incidence 4-coloring problem is polynomially solvable and the interval incidence 5-coloring problem is N P-complete, and they asked if χii(G) ≤ 2∆(G) holds for an arbitrary graph G. In this paper, we prove that an interval incidence 6-coloring always exists for any subcubic graph G with ∆(G) = 3.
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Similarities and Differences Between the Vertex Cover Number and the Weakly Connected Domination Number of a Graph
PublicationA vertex cover of a graph G = (V, E) is a set X ⊂ V such that each edge of G is incident to at least one vertex of X. The ve cardinality of a vertex cover of G. A dominating set D ⊆ V is a weakly connected dominating set of G if the subgraph G[D]w = (N[D], Ew) weakly induced by D, is connected, where Ew is the set of all edges having at least one vertex in D. The weakly connected domination number γw(G) of G is the minimum cardinality...
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Polynomial triset metric for unrooted phylogenetic trees
Publicationthe following paper presents a polynomial triset metric for unrooted phylogenetic trees (based on weighted bipartite graphs and the method of determining a minimum edge cover) and its basic characteristics. also a list of further directions of research and examples of the wider use of this metric is presented.
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Computational aspects of greedy partitioning of graphs
PublicationIn this paper we consider a variant of graph partitioning consisting in partitioning the vertex set of a graph into the minimum number of sets such that each of them induces a graph in hereditary class of graphs P (the problem is also known as P-coloring). We focus on the computational complexity of several problems related to greedy partitioning. In particular, we show that given a graph G and an integer k deciding if the greedy...
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Equitable and semi-equitable coloring of cubic graphs and its application in batch scheduling
PublicationIn the paper we consider the problems of equitable and semi-equitable coloring of vertices of cubic graphs. We show that in contrast to the equitable coloring, which is easy, the problem of semi-equitable coloring is NP- complete within a broad spectrum of graph parameters. This affects the complexity of batch scheduling of unit-length jobs with cubic incompatibility graph on three uniform processors to minimize...
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Cops, a fast robber and defensive domination on interval graphs
PublicationThe game of Cops and ∞-fast Robber is played by two players, one controlling c cops, the other one robber. The players alternate in turns: all the cops move at once to distance at most one each, the robber moves along any cop-free path. Cops win by sharing a vertex with the robber, the robber by avoiding capture indefinitely. The game was proposed with bounded robber speed by Fomin et al. in “Pursuing a fast robber on a graph”,...
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Three-fast-searchable graphs
PublicationIn the edge searching problem, searchers move from vertex to vertex in a graph to capture an invisible, fast intruder that may occupy either vertices or edges. Fast searching is a monotonic internal model in which, at every move, a new edge of the graph G must be guaranteed to be free of the intruder. That is, once all searchers are placed the graph G is cleared in exactly |E(G)| moves. Such a restriction obviously necessitates...
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Sprawiedliwe i półsprawiedliwe pokolorowania grafów kubicznych
PublicationW pracy rozpatrywane są sprawiedliwe i półsprawiedliwe pokolorowania grafów kubicznych. Pokazano, że w odróżnieniu od tego pierwszego, który jest łatwy, problem istnienia pokolorowań półsprawiedliwych jest NP-zupełny w szerokim zakresie parametrów grafów.
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The complexity of minimum-length path decompositions
PublicationWe consider a bi-criteria generalization of the pathwidth problem, where, for given integers k, l and a graph G, we ask whether there exists a path decomposition P of G such that the width of P is at most k and the number of bags in P, i.e., the length of P, is at most l. We provide a complete complexity classification of the problem in terms of k and l for general graphs. Contrary to the original pathwidth problem, which is fixed-parameter...
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On Computational Aspects of Greedy Partitioning of Graphs
PublicationIn this paper we consider a problem of graph P-coloring consisting in partitioning the vertex set of a graph such that each of the resulting sets induces a graph in a given additive, hereditary class of graphs P. We focus on partitions generated by the greedy algorithm. In particular, we show that given a graph G and an integer k deciding if the greedy algorithm outputs a P-coloring with a least k colors is NP-complete for an infinite...
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Global defensive sets in graphs
PublicationIn the paper we study a new problem of finding a minimum global defensive set in a graph which is a generalization of the global alliance problem. For a given graph G and a subset S of a vertex set of G, we define for every subset X of S the predicate SEC ( X ) = true if and only if | N [ X ] ∩ S | ≥ | N [ X ] \ S | holds, where N [ X ] is a closed neighbourhood of X in graph G. A set S is a defensive alliance if and only if for...