(H, k)-reachability in H-arc-colored digraphs
Let H be a digraph possibly with loops, D be a digraph, and k be an integer, k ≥ 3 . An H -coloring ζ is a map ζ : A ( D ) → V ( H ) . An ( H , k )-walk W in D is a walk W = ( x 0 , ⋯ , x n ) with length at most k such that ( ζ ( x 0 , x 1 ) , ⋯ , ζ ( x n - 1 , x n ) ) is a walk in H . An ( H , k...
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| Published in | Boletín de la Sociedad Matemática Mexicana Vol. 29; no. 1 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.03.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Summary: | Let
H
be a digraph possibly with loops,
D
be a digraph, and
k
be an integer,
k
≥
3
. An
H
-coloring
ζ
is a map
ζ
:
A
(
D
)
→
V
(
H
)
. An (
H
,
k
)-walk
W
in
D
is a walk
W
=
(
x
0
,
⋯
,
x
n
)
with length at most
k
such that
(
ζ
(
x
0
,
x
1
)
,
⋯
,
ζ
(
x
n
-
1
,
x
n
)
)
is a walk in
H
. An (
H
,
k
)-path in
D
is an (
H
,
k
)-walk which is a path in
D
. In this work, we introduce the reachability by (
H
,
k
)-paths as follows, for
u
,
v
∈
V
(
D
)
, we say that
u
reaches
v
by (
H
,
k
)-paths if there exists an (
H
,
k
)-path from
u
to
v
in
D
. Naturally, this new reachability concept can be used to model several connectivity problems. We focus on one of the many aspects of the reachability by (
H
,
k
)-paths, the (
H
,
k
)-kernels. A subset
N
of
V
(
D
) is an (
H
,
k
)-kernel if
N
is an (
H
,
k
)-independent (a subset
S
of
V
(
D
) such that no vertex in
S
can reach another (different) vertex in
S
by
(
H
,
k
-
1
)
-paths) and
(
H
,
k
-
1
)
-absorbent (a subset
S
of
V
(
D
) such that every vertex in
V
(
D
)
-
S
reaches some vertex in
S
by
(
H
,
k
-
1
)
-paths). A digraph
D
is (
H
,
k
)-path-quasi-transitive, if for every three vertices
x
,
y
and
w
of
D
such that there are an (
H
,
k
)-path from
x
to
y
and an (
H
,
k
)-path from
y
to
w
in
D
, then there is an (
H
,
k
)-path from
x
to
w
or an (
H
,
k
)-path from
w
to
x
in
D
. We give sufficient conditions for a
(
H
,
k
-
1
)
-path-quasi-transitive digraph that has an (
H
,
k
)-kernel. As a main result, we give sufficient conditions for a partition
ξ
of
V
(
H
) such that the arc set colored with the colors for every part of
ξ
induces an
(
H
,
k
-
1
)
-path-quasi-transitive digraph in
D
, to imply the existence of an (
H
,
k
)-kernel in
D
. This result generalizes the results of Casas-Bautista et al. (2015), and Hernández-Lorenzana and Sánchez-López (2022). Finally, we show two applications of (
H
,
k
)-kernels. |
|---|---|
| ISSN: | 1405-213X 2296-4495 |
| DOI: | 10.1007/s40590-022-00484-x |