231 research outputs found
Graph Treewidth and Geometric Thickness Parameters
Consider a drawing of a graph in the plane such that crossing edges are
coloured differently. The minimum number of colours, taken over all drawings of
, is the classical graph parameter "thickness". By restricting the edges to
be straight, we obtain the "geometric thickness". By further restricting the
vertices to be in convex position, we obtain the "book thickness". This paper
studies the relationship between these parameters and treewidth.
Our first main result states that for graphs of treewidth , the maximum
thickness and the maximum geometric thickness both equal .
This says that the lower bound for thickness can be matched by an upper bound,
even in the more restrictive geometric setting. Our second main result states
that for graphs of treewidth , the maximum book thickness equals if and equals if . This refutes a conjecture of Ganley and
Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved
for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of
the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in
Computer Science 3843:129-140, Springer, 2006. The full version was published
in Discrete & Computational Geometry 37(4):641-670, 2007. That version
contained a false conjecture, which is corrected on page 26 of this versio
Energy Complexity of Distance Computation in Multi-hop Networks
Energy efficiency is a critical issue for wireless devices operated under
stringent power constraint (e.g., battery). Following prior works, we measure
the energy cost of a device by its transceiver usage, and define the energy
complexity of an algorithm as the maximum number of time slots a device
transmits or listens, over all devices. In a recent paper of Chang et al. (PODC
2018), it was shown that broadcasting in a multi-hop network of unknown
topology can be done in energy. In this paper, we continue
this line of research, and investigate the energy complexity of other
fundamental graph problems in multi-hop networks. Our results are summarized as
follows.
1. To avoid spending energy, the broadcasting protocols of Chang
et al. (PODC 2018) do not send the message along a BFS tree, and it is open
whether BFS could be computed in energy, for sufficiently large . In
this paper we devise an algorithm that attains energy
cost.
2. We show that the framework of the round lower bound proof
for computing diameter in CONGEST of Abboud et al. (DISC 2017) can be adapted
to give an energy lower bound in the wireless network model
(with no message size constraint), and this lower bound applies to -arboricity graphs. From the upper bound side, we show that the energy
complexity of can be attained for bounded-genus graphs
(which includes planar graphs).
3. Our upper bounds for computing diameter can be extended to other graph
problems. We show that exact global minimum cut or approximate -- minimum
cut can be computed in energy for bounded-genus graphs
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
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