10,083 research outputs found
Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms
It is known that graphs on n vertices with minimum degree at least 3 have
spanning trees with at least n/4+2 leaves and that this can be improved to
(n+4)/3 for cubic graphs without the diamond K_4-e as a subgraph. We generalize
the second result by proving that every graph with minimum degree at least 3,
without diamonds and certain subgraphs called blossoms, has a spanning tree
with at least (n+4)/3 leaves, and generalize this further by allowing vertices
of lower degree. We show that it is necessary to exclude blossoms in order to
obtain a bound of the form n/3+c.
We use the new bound to obtain a simple FPT algorithm, which decides in
O(m)+O^*(6.75^k) time whether a graph of size m has a spanning tree with at
least k leaves. This improves the best known time complexity for MAX LEAF
SPANNING TREE.Comment: 25 pages, 27 Figure
Embedding nearly-spanning bounded degree trees
We derive a sufficient condition for a sparse graph G on n vertices to
contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n
vertices, in terms of the expansion properties of G. As a result we show that
for fixed d\geq 2 and 0<\epsilon<1, there exists a constant c=c(d,\epsilon)
such that a random graph G(n,c/n) contains almost surely a copy of every tree T
on (1-\epsilon)n vertices with maximum degree at most d. We also prove that if
an (n,D,\lambda)-graph G (i.e., a D-regular graph on n vertices all of whose
eigenvalues, except the first one, are at most \lambda in their absolute
values) has large enough spectral gap D/\lambda as a function of d and
\epsilon, then G has a copy of every tree T as above
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
- …