129 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
The Strong Nine Dragon Tree Conjecture is true for
The arboricity of an undirected graph is the minimal
number such that can be partitioned into forests.
Nash-Williams' formula states that , where
is the maximum of over all subgraphs of with .
The Strong Nine Dragon Tree Conjecture states that if for , then there is a partition of the
edge set of into forests such that one forest has at most edges
in each connected component.
We settle the conjecture for . For , we cannot
prove the conjecture, however we show that there exists a partition in which
the connected components in one forest have at most edges.
As an application of this theorem, we show that every -edge-connected
planar graph has a -thin spanning tree. This theorem is best
possible, in the sense that we cannot replace -edge-connected with
-edge-connected, even if we replace with any positive real
number less than . This strengthens a result of Merker and Postle which
showed -edge-connected planar graphs have a -thin spanning
tree.Comment: 24 pages, paper updated in accordance to referee comment
Digraph Colouring and Arc-Connectivity
The dichromatic number of a digraph is the minimum size of
a partition of its vertices into acyclic induced subgraphs. We denote by
the maximum local edge connectivity of a digraph . Neumann-Lara
proved that for every digraph , . In this
paper, we characterize the digraphs for which . This generalizes an analogue result for undirected graph proved by Stiebitz
and Toft as well as the directed version of Brooks' Theorem proved by Mohar.
Along the way, we introduce a generalization of Haj\'os join that gives a new
way to construct families of dicritical digraphs that is of independent
interest.Comment: 34 pages, 11 figure
Open Problems in (Hyper)Graph Decomposition
Large networks are useful in a wide range of applications. Sometimes problem
instances are composed of billions of entities. Decomposing and analyzing these
structures helps us gain new insights about our surroundings. Even if the final
application concerns a different problem (such as traversal, finding paths,
trees, and flows), decomposing large graphs is often an important subproblem
for complexity reduction or parallelization. This report is a summary of
discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph
Decomposition" and presents currently open problems and future directions in
the area of (hyper)graph decomposition
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Non-crossing shortest paths in planar graphs with applications to max flow, and path graphs
This thesis is concerned with non-crossing shortest paths in planar graphs with applications to st-max flow vitality and path graphs.
In the first part we deal with non-crossing shortest paths in a plane graph G, i.e., a planar graph with a fixed planar embedding, whose extremal vertices lie on the same face of G. The first two results are the computation of the lengths of the non-crossing shortest paths knowing their union, and the computation of the union in the unweighted case. Both results require linear time and we use them to describe an efficient algorithm able to give an additive guaranteed approximation of edge and vertex vitalities with respect to the st-max flow in undirected planar graphs, that is the max flow decrease when the edge/vertex is removed from the graph. Indeed, it is well-known that the st-max flow in an undirected planar graph can be reduced to a problem of non-crossing shortest paths in the dual graph. We conclude this part by showing that the union of non-crossing shortest paths in a plane graph can be covered with four forests so that each path is contained in at least one forest.
In the second part of the thesis we deal with path graphs and directed path graphs, where a (directed) path graph is the intersection graph of paths in a (directed) tree. We introduce a new characterization of path graphs that simplifies the existing ones in the literature. This characterization leads to a new list of local forbidden subgraphs of path graphs and to a new algorithm able to recognize path graphs and directed path graphs. This algorithm is more intuitive than the existing ones and does not require sophisticated data structures
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