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## Structure Theorem (structure + theorem)
## Selected Abstracts## The fractional matching numbers of graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2002Yan LiuAbstract A fractional matching of a graph G is a function f that assigns to each edge a number in [0, 1] such that, for each vertex v, , f(e) , 1, where the sum is taken over all edges incident to v. The fractional matching number of G is the supremum of ,e,E(G)f(e) over all fractional matchings f. In this paper, we provide a new formula for calculating the fractional matching numbers of graphs using the Gallai,Edmonds Structure Theorem. Thus, we characterize graphs for which the fractional matching number equals the matching number and graphs for which the fractional matching number is the maximum possible (one-half the number of vertices). © 2002 Wiley Periodicals, Inc. [source] ## A structure theorem for graphs with no cycle with a unique chord and its consequences JOURNAL OF GRAPH THEORY, Issue 1 2010Nicolas TrotignonAbstract We give a structural description of the class ,, of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in ,, is either in some simple basic class or has a decomposition. Basic classes are chordless cycles, cliques, bipartite graphs with one side containing only nodes of degree 2 and induced subgraphs of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for ,,, i.e. every graph in ,, can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations, and all graphs built this way are in ,,. This has several consequences: an ,,(nm) -time algorithm to decide whether a graph is in ,,, an ,,(n+ m) -time algorithm that finds a maximum clique of any graph in ,,, and an ,,(nm) -time coloring algorithm for graphs in ,,. We prove that every graph in ,, is either 3-colorable or has a coloring with , colors where , is the size of a largest clique. The problem of finding a maximum stable set for a graph in ,, is known to be NP-hard. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 31,67, 2010 [source] ## On the solutions of the Moisil,Théodoresco system MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2008Juan Bory ReyesAbstract A structure theorem is proved for the solutions to the Moisil,Théodoresco system in open subsets of ,3. Furthermore, it is shown that the Cauchy transform maps L2(,2, ,0, 2+) isomorphically onto H2(,+3, ,0, 3+), thus proving an elegant generalization to ,2 of the classical notion of an analytic signal on the real line. Copyright © 2008 John Wiley & Sons, Ltd. [source] |