6 research outputs found

    Note on Perfect Forests

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    A spanning subgraph FF of a graph GG is called perfect if FF is a forest, the degree dF(x)d_F(x) of each vertex xx in FF is odd, and each tree of FF is an induced subgraph of GG. We provide a short proof of the following theorem of A.D. Scott (Graphs & Combin., 2001): a connected graph GG contains a perfect forest if and only if GG has an even number of vertices

    Note on Perfect Forests in Digraphs

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    A spanning subgraph FF of a graph GG is called {\em perfect} if FF is a forest, the degree dF(x)d_F(x) of each vertex xx in FF is odd, and each tree of FF is an induced subgraph of GG. Alex Scott (Graphs \& Combin., 2001) proved that every connected graph GG contains a perfect forest if and only if GG has an even number of vertices. We consider four generalizations to directed graphs of the concept of a perfect forest. While the problem of existence of the most straightforward one is NP-hard, for the three others this problem is polynomial-time solvable. Moreover, every digraph with only one strong component contains a directed forest of each of these three generalization types. One of our results extends Scott's theorem to digraphs in a non-trivial way

    Two short proofs of the Perfect Forest Theorem

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    A perfect forest is a spanning forest of a connected graph GG, all of whose components are induced subgraphs of GG and such that all vertices have odd degree in the forest. A perfect forest generalised a perfect matching since, in a matching, all components are trees on one edge. Scott first proved the Perfect Forest Theorem, namely, that every connected graph of even order has a perfect forest. Gutin then gave another proof using linear algebra. We give here two very short proofs of the Perfect Forest Theorem which use only elementary notions from graph theory. Both our proofs yield polynomial-time algorithms for finding a perfect forest in a connected graph of even order

    Perfect Forests in Graphs and Their Extensions

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    Let G be a graph on n vertices. For i ? {0,1} and a connected graph G, a spanning forest F of G is called an i-perfect forest if every tree in F is an induced subgraph of G and exactly i vertices of F have even degree (including zero). An i-perfect forest of G is proper if it has no vertices of degree zero. Scott (2001) showed that every connected graph with even number of vertices contains a (proper) 0-perfect forest. We prove that one can find a 0-perfect forest with minimum number of edges in polynomial time, but it is NP-hard to obtain a 0-perfect forest with maximum number of edges. We also prove that for a prescribed edge e of G, it is NP-hard to obtain a 0-perfect forest containing e, but we can find a 0-perfect forest not containing e in polynomial time. It is easy to see that every graph with odd number of vertices has a 1-perfect forest. It is not the case for proper 1-perfect forests. We give a characterization of when a connected graph has a proper 1-perfect forest

    Maximum odd induced subgraph of a graph concerning its chromatic number

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    Let fo(G)f_{o}(G) be the maximum order of an odd induced subgraph of GG. In 1992, Scott proposed a conjecture that fo(G)≥n2χ(G)f_{o}(G)\geq \frac {n} {2\chi(G)} for a graph GG of order nn without isolated vertices, where χ(G)\chi(G) is the chromatic number of GG. In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang and Wargo in 1997, which states that fo(G)≥2⌊n4⌋f_{o}(G)\geq 2\lfloor\frac {n} {4}\rfloor for a connected graph GG of order nn. Scott's conjecture is open for a graph with chromatic number at least 3.Comment: 10 pages, 1 figur

    Note on Perfect Forests

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