4 research outputs found
On Panchromatic Patterns
Given D and H two digraphs, D is H-coloured iff the arcs of D are coloured
with the vertices of H. After defining what do we mean by an H-walk in the
coloured D, we characterise those H, which we call panchromatic patterns, for
which all D and all H-colourings of D admit a kernel by H-walks. This solves a
problem of Arpin and Linek from 2007
A dichotomy for the kernel by -walks problem in digraphs
Let be a digraph which may contain loops, and let be a loopless digraph with a coloring of its arcs . An
-walk of is a walk of such that is an arc of , for every . For , we say that reaches by -walks if there exists an
-walk from to in . A subset is a kernel by
-walks of if every vertex in reaches by -walks some
vertex in , and no vertex in can reach another vertex in by
-walks.
A panchromatic pattern is a digraph such that every arc-colored digraph
has a kernel by -walks. In this work, we prove that every digraph is
either a panchromatic pattern, or the problem of determining whether an
arc-colored digraph has a kernel by -walks is -complete.Comment: 20 pages, 3 figure
Panchromatic patterns by paths
Let be a digraph, possibly with loops, and let
be a loopless multidigraph with a colouring of its arcs . An -path of is a path of such that
is an arc of for every . For , we say that reaches by -paths if there
exists an -path from to in . A subset is
-absorbent of if every vertex in reaches by -paths some
vertex in , and it is -independent if no vertex in can reach another
(different) vertex in by -pahts. An -kernel is an independent by
-paths and absorbent by -paths subset of .
We define as the set of digraphs such that any
-arc-coloured tournament has an -absorbent by paths vertex; the set
consists of the digraphs such that any
-arc-coloured digraph has an independent, -absorbent by paths set;
analogously, the set is the set of digraphs such
that every -arc-coloured digraph contains an -kernel by paths.
In this work, we present a characterization of , and
provide structural properties of the digraphs in which
settle up its characterization except for the analysis of a single digraph on
three vertices.Comment: 27 pages, 9 figure
H-Kernels by Walks
We prove that, if every cycle of is an -cycle, then has an
-kernel by walks