31 research outputs found
Acyclic homomorphisms to stars of graph Cartesian products and chordal bipartite graphs
AbstractHomomorphisms to a given graph H (H-colourings) are considered in the literature among other graph colouring concepts. We restrict our attention to a special class of H-colourings, namely H is assumed to be a star. Our additional requirement is that the set of vertices of a graph G mapped into the central vertex of the star and any other colour class induce in G an acyclic subgraph. We investigate the existence of such a homomorphism to a star of given order. The complexity of this problem is studied. Moreover, the smallest order of a star for which a homomorphism of a given graph G with desired features exists is considered. Some exact values and many bounds of this number for chordal bipartite graphs, cylinders, grids, in particular hypercubes, are given. As an application of these results, we obtain some bounds on the cardinality of the minimum feedback vertex set for specified graph classes
Rectangle Visibility Numbers of Graphs
Very-Large Scale Integration (VLSI) is the problem of arranging components on the surface of a circuit board and developing the wired network between components. One methodology in VLSI is to treat the entire network as a graph, where the components correspond to vertices and the wired connections correspond to edges. We say that a graph G has a rectangle visibility representation if we can assign each vertex of G to a unique axis-aligned rectangle in the plane such that two vertices u and v are adjacent if and only if there exists an unobstructed horizontal or vertical channel of finite width between the two rectangles that correspond to u and v. If G has such a representation, then we say that G is a rectangle visibility graph.
Since it is likely that multiple components on a circuit board may represent the same electrical node, we may consider implementing this idea with rectangle visibility graphs. The rectangle visibility number of a graph G, denoted r(G), is the minimum k such that G has a rectangle visibility representation in which each vertex of G corresponds to at most k rectangles. In this thesis, we prove results on rectangle visibility numbers of trees, complete graphs, complete bipartite graphs, and (1,n)-hilly graphs, which are graphs where there is no path of length 1 between vertices of degree n or more
Three ways to cover a graph
We consider the problem of covering an input graph with graphs from a
fixed covering class . The classical covering number of with respect to
is the minimum number of graphs from needed to cover the edges of
without covering non-edges of . We introduce a unifying notion of three
covering parameters with respect to , two of which are novel concepts only
considered in special cases before: the local and the folded covering number.
Each parameter measures "how far'' is from in a different way. Whereas
the folded covering number has been investigated thoroughly for some covering
classes, e.g., interval graphs and planar graphs, the local covering number has
received little attention.
We provide new bounds on each covering number with respect to the following
covering classes: linear forests, star forests, caterpillar forests, and
interval graphs. The classical graph parameters that result this way are
interval number, track number, linear arboricity, star arboricity, and
caterpillar arboricity. As input graphs we consider graphs of bounded
degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as
well as outerplanar, planar bipartite, and planar graphs. For several pairs of
an input class and a covering class we determine exactly the maximum ordinary,
local, and folded covering number of an input graph with respect to that
covering class.Comment: 20 pages, 4 figure
Constructing disjoint Steiner trees in Sierpi\'{n}ski graphs
Let be a graph and with . Then the trees
in are \emph{internally disjoint Steiner trees}
connecting (or -Steiner trees) if and
for every pair of distinct integers , . Similarly, if we only have the condition but without the condition , then they are
\emph{edge-disjoint Steiner trees}. The \emph{generalized -connectivity},
denoted by , of a graph , is defined as
,
where is the maximum number of internally disjoint -Steiner
trees. The \emph{generalized local edge-connectivity} is the
maximum number of edge-disjoint Steiner trees connecting in . The {\it
generalized -edge-connectivity} of is defined as
. These
measures are generalizations of the concepts of connectivity and
edge-connectivity, and they and can be used as measures of vulnerability of
networks. It is, in general, difficult to compute these generalized
connectivities. However, there are precise results for some special classes of
graphs. In this paper, we obtain the exact value of
for , and the exact value of for
, where is the Sierpi\'{n}ski graphs with order
. As a direct consequence, these graphs provide additional interesting
examples when . We also study the
some network properties of Sierpi\'{n}ski graphs