22 research outputs found
Directed Minors III. Directed Linked Decompositions
Thomas proved that every undirected graph admits a linked tree decomposition
of width equal to its treewidth. In this paper, we generalize Thomas's theorem
to digraphs. We prove that every digraph G admits a linked directed path
decomposition and a linked DAG decomposition of width equal to its directed
pathwidth and DAG-width respectively
Hitting minors, subdivisions, and immersions in tournaments
The Erd\H{o}s-P\'osa property relates parameters of covering and packing of
combinatorial structures and has been mostly studied in the setting of
undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim,
and Seymour to show that, for every directed graph (resp.
strongly-connected directed graph ), the class of directed graphs that
contain as a strong minor (resp. butterfly minor, topological minor) has
the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove
that if is a strongly-connected directed graph, the class of directed
graphs containing as an immersion has the edge-Erd\H{o}s-P\'osa property in
the class of tournaments.Comment: Accepted to Discrete Mathematics & Theoretical Computer Science.
Difference with the previous version: use of the DMTCS article class. For a
version with hyperlinks see the previous versio
Forbidden Directed Minors and Kelly-width
Partial 1-trees are undirected graphs of treewidth at most one. Similarly,
partial 1-DAGs are directed graphs of KellyWidth at most two. It is well-known
that an undirected graph is a partial 1-tree if and only if it has no K_3
minor. In this paper, we generalize this characterization to partial 1-DAGs. We
show that partial 1-DAGs are characterized by three forbidden directed minors,
K_3, N_4 and M_5
On the pathwidth of almost semicomplete digraphs
We call a digraph {\em -semicomplete} if each vertex of the digraph has at
most non-neighbors, where a non-neighbor of a vertex is a vertex such that there is no edge between and in either direction.
This notion generalizes that of semicomplete digraphs which are
-semicomplete and tournaments which are semicomplete and have no
anti-parallel pairs of edges. Our results in this paper are as follows. (1) We
give an algorithm which, given an -semicomplete digraph on vertices
and a positive integer , in time either
constructs a path-decomposition of of width at most or concludes
correctly that the pathwidth of is larger than . (2) We show that there
is a function such that every -semicomplete digraph of pathwidth
at least has a semicomplete subgraph of pathwidth at least .
One consequence of these results is that the problem of deciding if a fixed
digraph is topologically contained in a given -semicomplete digraph
admits a polynomial-time algorithm for fixed .Comment: 33pages, a shorter version to appear in ESA 201