58,851 research outputs found
-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum
A -WORM coloring of a graph is an assignment of colors to the
vertices in such a way that the vertices of each -subgraph of get
precisely two colors. We study graphs which admit at least one such
coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014)
161--173] who asked whether every such graph has a -WORM coloring with two
colors. In fact for every integer there exists a -WORM colorable
graph in which the minimum number of colors is exactly . There also exist
-WORM colorable graphs which have a -WORM coloring with two colors
and also with colors but no coloring with any of colors. We
also prove that it is NP-hard to determine the minimum number of colors and
NP-complete to decide -colorability for every (and remains
intractable even for graphs of maximum degree 9 if ). On the other hand,
we prove positive results for -degenerate graphs with small , also
including planar graphs. Moreover we point out a fundamental connection with
the theory of the colorings of mixed hypergraphs. We list many open problems at
the end.Comment: 18 page
Parameterized Complexity of Simultaneous Planarity
Given input graphs , where each pair , with
shares the same graph , the problem Simultaneous Embedding With
Fixed Edges (SEFE) asks whether there exists a planar drawing for each input
graph such that all drawings coincide on . While SEFE is still open for the
case of two input graphs, the problem is NP-complete for [Schaefer,
JGAA 13]. In this work, we explore the parameterized complexity of SEFE. We
show that SEFE is FPT with respect to plus the vertex cover number or the
feedback edge set number of the the union graph . Regarding the shared graph , we show that SEFE is NP-complete, even if
is a tree with maximum degree 4. Together with a known NP-hardness
reduction [Angelini et al., TCS 15], this allows us to conclude that several
parameters of , including the maximum degree, the maximum number of degree-1
neighbors, the vertex cover number, and the number of cutvertices are
intractable. We also settle the tractability of all pairs of these parameters.
We give FPT algorithms for the vertex cover number plus either of the first two
parameters and for the number of cutvertices plus the maximum degree, whereas
we prove all remaining combinations to be intractable.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
On the Complexity of Two Dots for Narrow Boards and Few Colors
Two Dots is a popular single-player puzzle video game for iOS and Android. A level of this game consists of a grid of colored dots. The player connects two or more adjacent dots, removing them from the grid and causing the remaining dots to fall, as if influenced by gravity. One special move, which is frequently a game-changer, consists of connecting a cycle of dots: this removes all the dots of the given color from the grid. The goal is to remove a certain number of dots of each color using a limited number of moves. The computational complexity of Two Dots has already been addressed in [Misra, FUN 2016], where it has been shown that the general version of the problem is NP-complete. Unfortunately, the known reductions produce Two Dots levels having both a large number of colors and many columns. This does not completely match the spirit of the game, where, on the one hand, only few colors are allowed, and on the other hand, the grid of the game has only a constant number of columns. In this paper, we partially fill this gap by assessing the computational complexity of Two Dots instances having a small number of colors or columns. More precisely, we show that Two Dots is hard even for instances involving only 3 colors or 2 columns. As a contrast, we also prove that the problem can be solved in polynomial-time on single-column instances with a constant number of goals
Strings-and-Coins and Nimstring are PSPACE-complete
We prove that Strings-and-Coins -- the combinatorial two-player game
generalizing the dual of Dots-and-Boxes -- is strongly PSPACE-complete on
multigraphs. This result improves the best previous result, NP-hardness, argued
in Winning Ways. Our result also applies to the Nimstring variant, where the
winner is determined by normal play; indeed, one step in our reduction is the
standard reduction (also from Winning Ways) from Nimstring to
Strings-and-Coins.Comment: 10 pages, 7 figures. Improved wording and figures; cite
arXiv:2105.0283
Global offensive -alliances in digraphs
In this paper, we initiate the study of global offensive -alliances in
digraphs. Given a digraph , a global offensive -alliance in a
digraph is a subset such that every vertex outside of
has at least one in-neighbor from and also at least more in-neighbors
from than from outside of , by assuming is an integer lying between
two minus the maximum in-degree of and the maximum in-degree of . The
global offensive -alliance number is the minimum
cardinality among all global offensive -alliances in . In this article we
begin the study of the global offensive -alliance number of digraphs. For
instance, we prove that finding the global offensive -alliance number of
digraphs is an NP-hard problem for any value and that it remains NP-complete even when
restricted to bipartite digraphs when we consider the non-negative values of
given in the interval above. Based on these facts, lower bounds on
with characterizations of all digraphs attaining the bounds
are given in this work. We also bound this parameter for bipartite digraphs
from above. For the particular case , an immediate result from the
definition shows that for all digraphs ,
in which stands for the domination number of . We show that
these two digraph parameters are the same for some infinite families of
digraphs like rooted trees and contrafunctional digraphs. Moreover, we show
that the difference between and can be
arbitrary large for directed trees and connected functional digraphs
Partitioning a graph into degenerate subgraphs
Let be a connected graph with maximum degree distinct
from . Given integers and , is
said to be -partitionable if there exists a partition of
into sets~ such that is -degenerate for
. In this paper, we prove that we can find a -partition of in -time whenever and . This generalizes a result of Bonamy
et al. (MFCS, 2017) and can be viewed as an algorithmic extension of Brooks'
theorem and several results on vertex arboricity of graphs of bounded maximum
degree.
We also prove that deciding whether is -partitionable is
-complete for every and pairs of non-negative integers
such that and . This resolves an
open problem of Bonamy et al. (manuscript, 2017). Combined with results of
Borodin, Kostochka and Toft (\emph{Discrete Mathematics}, 2000), Yang and Yuan
(\emph{Discrete Mathematics}, 2006) and Wu, Yuan and Zhao (\emph{Journal of
Mathematical Study}, 1996), it also settles the complexity of deciding whether
a graph with bounded maximum degree can be partitioned into two subgraphs of
prescribed degeneracy.Comment: 16 pages; minor revisio
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