Article in volume
Authors:
Hortensia Galeana-Sánchez
Instituto de Matemáticas, UNAMUniversidad Nacional Autónoma de MéxicoCiudad Universitaria04510, México, D.F.MEXICO
email: hgaleana@matem.unam.mx
Title:
$(k,H)$-kernels in nearly tournaments
PDFSource:
Discussiones Mathematicae Graph Theory 44(2) (2024) 639-662
Received: 2021-09-08 , Revised: 2022-06-16 , Accepted: 2022-06-17 , Available online: 2022-07-13 , https://doi.org/10.7151/dmgt.2463
Abstract:
Let $H$ be a digraph, possibly with loops, $D$ a digraph without loops, and
$\rho : A(D) \rightarrow V(H)$ a coloring of $A(D)$ ($D$ is said to be an
$H$-colored digraph). If $W=(x_{0}, \ldots , x_{n})$ is a walk in $D$, and
$i \in \{ 0, \ldots , n-1 \}$, then we say that there is an obstruction on $x_{i}$
whenever $(\rho(x_{i-1}, x_{i}), \rho (x_{i}, x_{i+1})) \notin A(H)$ (when
$x_{0} = x_{n}$ the indices are taken modulo $n$). We denote by $O_{H}(W)$ the
set $\{ i \in \{0, \ldots , n-1 \} :$ there is an obstruction on $x_{i} \}$.
The $H$-length of $W$, denoted by $l_{H}(W)$, is defined by $|O_{H}(W)|$ if
$W$ is closed or $|O_{H}(W)|+1$ in the other case.
A $(k, H)$-kernel of an $H$-colored digraph $D$ ($k \geq 2$) is a subset of
vertices of $D$, say $S$, such that, for every pair of different vertices in
$S$, every path between them has $H$-length at least $k$, and for every vertex
$x \in V(D) \setminus S$ there exists an $xS$-path with $H$-length at most $k-1$.
This concept widely generalize previous nice concepts such as kernel, $k$-kernel,
kernel by monochromatic paths, kernel by properly colored paths, and $H$-kernel.
In this paper, we introduce the concept of $(k,H)$-kernel and we will study the
existence of $(k,H)$-kernels in interesting classes of digraphs, called nearly
tournaments, which have been large and widely studied due to its applications and
theoretical results. We will show several conditions that guarantee the
existence of a $(k,H)$-kernel in tournaments, $r$-transitive digraphs,
$r$-quasi-transitive digraphs, multipartite tournaments, and local tournaments.
As a consequence, previous results for $k$-kernels and kernels by alternating
paths will be generalized, and some conditions for the existence of kernels by
monochromatic paths and $H$-kernels in nearly tournaments will be shown.
Keywords:
kernel, $k$-kernel, $H$-kernel, $H$-coloring, kernel by monochromatic paths, kernel by alternating paths
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