112 research outputs found
The degree-diameter problem for sparse graph classes
The degree-diameter problem asks for the maximum number of vertices in a
graph with maximum degree and diameter . For fixed , the answer
is . We consider the degree-diameter problem for particular
classes of sparse graphs, and establish the following results. For graphs of
bounded average degree the answer is , and for graphs of
bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases
for fixed . For graphs of given treewidth, we determine the the maximum
number of vertices up to a constant factor. More precise bounds are given for
graphs of given treewidth, graphs embeddable on a given surface, and
apex-minor-free graphs
Edge Partitions of Optimal -plane and -plane Graphs
A topological graph is a graph drawn in the plane. A topological graph is
-plane, , if each edge is crossed at most times. We study the
problem of partitioning the edges of a -plane graph such that each partite
set forms a graph with a simpler structure. While this problem has been studied
for , we focus on optimal -plane and -plane graphs, which are
-plane and -plane graphs with maximum density. We prove the following
results. (i) It is not possible to partition the edges of a simple optimal
-plane graph into a -plane graph and a forest, while (ii) an edge
partition formed by a -plane graph and two plane forests always exists and
can be computed in linear time. (iii) We describe efficient algorithms to
partition the edges of a simple optimal -plane graph into a -plane graph
and a plane graph with maximum vertex degree , or with maximum vertex
degree if the optimal -plane graph is such that its crossing-free edges
form a graph with no separating triangles. (iv) We exhibit an infinite family
of simple optimal -plane graphs such that in any edge partition composed of
a -plane graph and a plane graph, the plane graph has maximum vertex degree
at least and the -plane graph has maximum vertex degree at least .
(v) We show that every optimal -plane graph whose crossing-free edges form a
biconnected graph can be decomposed, in linear time, into a -plane graph and
two plane forests
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