135,066 research outputs found
ON THE ROOTS OF EDGE COVER POLYNOMIALS OF GRAPHS
AbstractLet G be a simple graph of order n and size m. An edge covering of the graph G is a set of edges such that every vertex of the graph is incident to at least one edge of the set. Let e(G,k) be the number of edge covering sets of G of size k. The edge cover polynomial of G is the polynomial E(G,x)=∑k=1me(G,k)xk. In this paper, we obtain some results on the roots of the edge cover polynomials. We show that for every graph G with no isolated vertex, all the roots of E(G,x) are in the ball {z∈C:|z|<(2+3)21+3≃5.099}. We prove that if every block of the graph G is K2 or a cycle, then all real roots of E(G,x) are in the interval (−4,0]. We also show that for every tree T of order n we have ξR(K1,n−1)≤ξR(T)≤ξR(Pn), where −ξR(T) is the smallest real root of E(T,x), and Pn,K1,n−1 are the path and the star of order n, respectively
Gamma-Set Domination Graphs. I: Complete Biorientations of \u3cem\u3eq-\u3c/em\u3eExtended Stars and Wounded Spider Graphs
The domination number of a graph G, γ(G), and the domination graph of a digraph D, dom(D) are integrated in this paper. The γ-set domination graph of the complete biorientation of a graph G, domγ(G) is created. All γ-sets of specific trees T are found, and dom-γ(T) is characterized for those classes
Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices
We study the Steiner Tree problem, in which a set of terminal vertices needs
to be connected in the cheapest possible way in an edge-weighted graph. This
problem has been extensively studied from the viewpoint of approximation and
also parametrization. In particular, on one hand Steiner Tree is known to be
APX-hard, and W[2]-hard on the other, if parameterized by the number of
non-terminals (Steiner vertices) in the optimum solution. In contrast to this
we give an efficient parameterized approximation scheme (EPAS), which
circumvents both hardness results. Moreover, our methods imply the existence of
a polynomial size approximate kernelization scheme (PSAKS) for the considered
parameter.
We further study the parameterized approximability of other variants of
Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of
these an EPAS is likely to exist for the studied parameter: for Steiner Forest
an easy observation shows that the problem is APX-hard, even if the input graph
contains no Steiner vertices. For Directed Steiner Tree we prove that
approximating within any function of the studied parameter is W[1]-hard.
Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree,
but a PSAKS does not. We also prove that there is an EPAS and a PSAKS for
Steiner Forest if in addition to the number of Steiner vertices, the number of
connected components of an optimal solution is considered to be a parameter.Comment: 23 pages, 6 figures An extended abstract appeared in proceedings of
STACS 201
Spanning trees without adjacent vertices of degree 2
Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist
highly connected graphs in which every spanning tree contains vertices of
degree 2. Using a result of Alon and Wormald, we show that there exists a
natural number such that every graph of minimum degree at least
contains a spanning tree without adjacent vertices of degree 2. Moreover, we
prove that every graph with minimum degree at least 3 has a spanning tree
without three consecutive vertices of degree 2
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