We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a (1±ϵ)-approximation in O(nω/t...
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A graph G is said to be Ramsey size-linear if r(G, H) = OG(e(H)) for every graph H with no isolated vertices. Erdős, Faudree, Rousseau, and Schelp observed that K4 is not Ramsey size-linear, but each of its proper su...
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We present an algorithm for verifying the local unitary (LU) equivalence of graph and stabilizer states. Our approach reduces the problem to solving a system of linear equations in modular arithmetic. Furthermore, we ...
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In this paper, we consider the (weighted) one-center problem of uncertain points on a cactus graph. Given are a cactus graph G and a set of n uncertain points. Each uncertain point has m possible locations on G with p...
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The classic greedy coloring (first-fit) algorithm considers the vertices of an input graph G in a given order and assigns the first available color to each vertex v in G. In the Grundy Coloring problem, the task is to...
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The classic greedy coloring (first-fit) algorithm considers the vertices of an input graph G in a given order and assigns the first available color to each vertex v in G. In the Grundy Coloring problem, the task is to find an ordering of the vertices that will force the greedy algorithm to use as many colors as possible. In the Partial Grundy Coloring, the task is also to color the graph using as many colors as possible. This time, however, we may select both the ordering in which the vertices are considered and which color to assign the vertex. The only constraint is that the color assigned to a vertex v is a color previously used for another vertex if such a color is available. Partial Grundy coloring of a graph corresponds to the vertex coloring produced by the greedy coloring heuristics called last-fit coloring, which assigns the last available color to each vertex in the given order. Whether Grundy Coloring and Partial Grundy Coloring admit fixed-parameter tractable (FPT) algorithms, algorithms with running time f(k)nO(1), where k is the number of colors, was posed as an open problem by Zaker and by Effantin et al., respectively. Recently, Aboulker et al. (STACS 2020 and Algorithmica 2022) resolved the question for Grundy Coloring in the negative by showing that the problem is W[1]-hard. For Partial Grundy Coloring, they obtain an FPT algorithm on graphs that do not contain Ki,j as a subgraph (a.k.a. Ki,j-free graphs). Aboulker et al. re-iterate the question of whether there exists an FPT algorithm for Partial Grundy Coloring on general graphs and also asks whether Grundy Coloring admits an FPT algorithm on Ki,j-free graphs. We give FPT algorithms for Partial Grundy Coloring on general graphs and for Grundy Coloring on Ki,j-free graphs, resolving both the questions in the affirmative. We believe that our new structural theorems for partial Grundy coloring and "representative-family" like sets for Ki,j-free graphs that we use in obtaining our results may have wi
In this paper we show that convex bipartite graphs can be decomposed into induced subgraphs consisting of biconvex connected components with no neighbourhood containment. We use this result to enumerate blue dominatin...
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Let G = (V,E) be a graph. The first Zagreb index of a graph G is defined as Pu∈V d2(u), where d(u) is the degree of vertex u in G. Using the Pólya-Szegő inequality, we in this paper present the first Zagreb ind...
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We revisit the problem of algorithmically deciding whether a given infinite connected graph has an Eulerian path, namely, a path that uses every edge exactly once. It has been recently observed that this problem is D3...
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We study algorithms for drawing planar graphs and 1-planar graphs using cubic Bézier curves with bounded curvature. We show that any n-vertex 1-planar graph has a 1-planar RAC drawing using a single cubic Bé...
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Real-world data is typically a noisy manifestation of a core pattern (schema), and the purpose of data mining algorithms is to uncover that pattern, thereby splitting (i.e. decomposing) the data into schema and noise....
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