The problems of recognizing series-parallel graphs, outerplanar graphs, and generalized series-parallel graphs have been studied separately in the past. Efficient algorithms have been presented. However, none of the a...
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The problems of recognizing series-parallel graphs, outerplanar graphs, and generalized series-parallel graphs have been studied separately in the past. Efficient algorithms have been presented. However, none of the algorithms are certifying. A certifying algorithm generates, in addition to its answer, a certificate that can be used by a checker (a separate algorithm) to verify the correctness of the answer. The certificate is positive if the answer is 'yes', and is negative if the answer is 'no'. In this paper, an O(|E| + |V |)-time certifying algorithm that simultaneously determines if a graph G = (V, E) is series- parallel, outerplanar, or generalized series-parallel is presented. The positive certificates are a construction sequence for constructing G if G is series-parallel, a generalized construction sequence for constructing G if G is generalized series-parallel but not series-parallel, and the edge set of the exterior boundary of an outerplanar embedding of G if G is outerplanar. The negative certificates are forbidden subgraphs or forbidden structures of G. All these certificates are generated by making only one pass over G after a preprocessing step decomposing G into its biconnected components.(c) 2022 Elsevier B.V. All rights reserved.
A series-parallel graph is a graph that does not contain a complete graph with four vertices as a minor. A new explicit simpler formula for the number of labeled series-parallel biconnected graphs with a given number ...
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Given a weighted graph (G, l) and its associated moduli space M(G, l), then a sufficient condition is provided which ensures that M(G, l) is a smooth manifold whenever G is a series-parallel graph. (C) 2014 Elsevier B...
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Given a weighted graph (G, l) and its associated moduli space M(G, l), then a sufficient condition is provided which ensures that M(G, l) is a smooth manifold whenever G is a series-parallel graph. (C) 2014 Elsevier B.V. All rights reserved.
Given a graph G = (V, E) and an integer k >= 1, the graph H = (V, F), where F is a family of elements (with repetitions allowed) of E, is a k-edge-connected spanning subgraph of G if H cannot be disconnected by del...
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Given a graph G = (V, E) and an integer k >= 1, the graph H = (V, F), where F is a family of elements (with repetitions allowed) of E, is a k-edge-connected spanning subgraph of G if H cannot be disconnected by deleting any k - 1 elements of F. The convex hull of incidence vectors of the k-edge-connected subgraphs of a graph G forms the k-edge-connected subgraph polyhedron of G. We prove that this polyhedron is box-totally dual integral if and only if G is series-parallel. In this case, we also provide an integer box-totally dual integral system describing this polyhedron.
It is known that any signed series-parallel graph (G, sigma) has circular chromatic number at most 10/3. This paper proves that for each rational r is an element of [2,10/3], there is a signed series-parallel graph (G...
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It is known that any signed series-parallel graph (G, sigma) has circular chromatic number at most 10/3. This paper proves that for each rational r is an element of [2,10/3], there is a signed series-parallel graph (G, sigma) whose circular chromatic number equals r. (C) 2021 Elsevier B.V. All rights reserved.
A series-parallel graph is a graph that does not contain a complete graph with four vertices as a minor. We find an asymptotics for the number of labeled connected series-parallel tetracyclic graphs with a large numbe...
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This paper investigates relationship between algebraic expressions and graphs. Our intent is to simplify graph expressions and eventually find their shortest representations. We prove the monotonicity results allowing...
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This paper investigates relationship between algebraic expressions and graphs. Our intent is to simplify graph expressions and eventually find their shortest representations. We prove the monotonicity results allowing to assert that the length of a shortest expression of any subgraph of a given graph is not greater than the length of a shortest expression of the graph. We describe the decomposition method for generating expressions of complete st-dags (two-terminal directed acyclic graphs) and estimate the ( ) corresponding expression complexities. Using these findings, we present an 2(O(log2 n)) upper bound for the length of a shortest expression for every n-vertex st-dag. (C) 2022 Elsevier B.V. All rights reserved.
A series-parallel graph is a graph that does not contain a complete graph with four vertices as a minor. An explicit formula for the number of labeled series-parallel tricyclic graphs with a given number of vertices i...
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The minimum color-degree perfectb-matching problem (Col-BM) is a new extension of the perfectb-matching problem to edge-colored graphs. The objective of Col-BM is to minimize the maximum number of differently colored ...
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The minimum color-degree perfectb-matching problem (Col-BM) is a new extension of the perfectb-matching problem to edge-colored graphs. The objective of Col-BM is to minimize the maximum number of differently colored edges in a perfectb-matching that are incident to the same node. We show that Col-BM isNP-hard on bipartite graphs by a reduction from (3,B2)-Sat, and conclude that there exists no(2 - epsilon)-approximation algorithm unlessP=NP. However, we identify a class of two-colored complete bipartite graphs on which we can solve Col-BM in polynomial time. Furthermore, we use dynamic programming to devise polynomial-time algorithms solving Col-BM with a fixed number of colors on series-parallel graphs and simple graphs with bounded treewidth.
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