689 research outputs found
On twin edge colorings in m-ary trees
Let k ≥ 2 be an integer and G be a connected graph of order at least 3. A twin k-edge coloring of G is a proper edge coloring of G that uses colors from ℤk and that induces a proper vertex coloring on G where the color of a vertex v is the sum (in ℤk) of the colors of the edges incident with v. The smallest integer k for which G has a twin k-edge coloring is the twin chromatic index of G and is denoted by χ′t(G). In this paper, we study the twin edge colorings in m-ary trees for m ≥ 2; in particular, the twin chromatic indexes of full m-ary trees that are not stars, r-regular trees for even r ≥ 2, and generalized star graphs that are not paths nor stars are completely determined. Moreover, our results confirm the conjecture that χ′t(G)≤Δ(G)+2 for every connected graph G (except C5) of order at least 3, for all trees of order at least 3
Twin chromatic indices of some graphs with maximum degree 3
Let k ≥ 2 be an integer and G be a connected graph of order at least 3. A twin k-edge coloring of G is a proper edge coloring of G that uses colors from k and that induces a proper vertex coloring on G where the color of a vertex v is the sum (in k ) of the colors of the edges incident with v. The smallest integer k for which G has a twin k-edge coloring is the twin chromatic index of G and is denoted by . In this paper, we determine the twin chromatic indices of circulant graphs , and some generalized Petersen graphs such as GP(3s, k), GP(m, 2), and GP(4s, l) where n ≥ 6 and n ≡ 0 (mod 4), s ≥ 1, k ≢ 0 (mod 3), m ≥ 3 and m {4, 5}, and l is odd. Moreover, we provide some sufficient conditions for a connected graph with maximum degree 3 to have twin chromatic index greater than 3
Parameterized Complexity of Fair Vertex Evaluation Problems
A prototypical graph problem is centered around a graph-theoretic property for a set of vertices and a solution to it is a set of vertices for which the desired property holds. The task is to decide whether, in the given graph, there exists a solution of a certain quality, where we use size as a quality measure. In this work, we are changing the measure to the fair measure (cf. Lin and Sahni [Li-Shin Lin and Sartaj Sahni, 1989]). The fair measure of a set of vertices S is (at most) k if the number of neighbors in the set S of any vertex (in the input graph) does not exceed k. One possible way to study graph problems is by defining the property in a certain logic. For a given objective, an evaluation problem is to find a set (of vertices) that simultaneously minimizes the assumed measure and satisfies an appropriate formula. More formally, we study the {MSO} Fair Vertex Evaluation, where the graph-theoretic property is described by an {MSO} formula.
In the presented paper we show that there is an FPT algorithm for the {MSO} Fair Vertex Evaluation problem for formulas with one free variable parameterized by the twin cover number of the input graph and the size of the formula. One may define an extended variant of {MSO} Fair Vertex Evaluation for formulas with l free variables; here we measure a maximum number of neighbors in each of the l sets. However, such variant is {W[1]}-hard for parameter l even on graphs with twin cover one.
Furthermore, we study the Fair Vertex Cover (Fair VC) problem. Fair VC is among the simplest problems with respect to the demanded property (i.e., the rest forms an edgeless graph). On the negative side, Fair VC is {W[1]}-hard when parameterized by both treedepth and feedback vertex set of the input graph. On the positive side, we provide an FPT algorithm for the parameter modular width
Geometric Graphs with Unbounded Flip-Width
We consider the flip-width of geometric graphs, a notion of graph width
recently introduced by Toru\'nczyk. We prove that many different types of
geometric graphs have unbounded flip-width. These include interval graphs,
permutation graphs, circle graphs, intersection graphs of axis-aligned line
segments or axis-aligned unit squares, unit distance graphs, unit disk graphs,
visibility graphs of simple polygons, -skeletons, 4-polytopes, rectangle
of influence graphs, and 3d Delaunay triangulations.Comment: 10 pages, 7 figures. To appear at CCCG 202
Stack-number is not bounded by queue-number
We describe a family of graphs with queue-number at most 4 but unbounded
stack-number. This resolves open problems of Heath, Leighton and Rosenberg
(1992) and Blankenship and Oporowski (1999)
Dynamic programming on bipartite tree decompositions
We revisit a graph width parameter that we dub bipartite treewidth, along
with its associated graph decomposition that we call bipartite tree
decomposition. Bipartite treewidth can be seen as a common generalization of
treewidth and the odd cycle transversal number. Intuitively, a bipartite tree
decomposition is a tree decomposition whose bags induce almost bipartite graphs
and whose adhesions contain at most one vertex from the bipartite part of any
other bag, while the width of such decomposition measures how far the bags are
from being bipartite. Adapted from a tree decomposition originally defined by
Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by
Tazari [Th. Comp. Sci. 2012], bipartite treewidth appears to play a crucial
role for solving problems related to odd-minors, which have recently attracted
considerable attention. As a first step toward a theory for solving these
problems efficiently, the main goal of this paper is to develop dynamic
programming techniques to solve problems on graphs of small bipartite
treewidth. For such graphs, we provide a number of para-NP-completeness
results, FPT-algorithms, and XP-algorithms, as well as several open problems.
In particular, we show that -Subgraph-Cover, Weighted Vertex
Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are
parameterized by bipartite treewidth. We provide the following complexity
dichotomy when is a 2-connected graph, for each of -Subgraph-Packing,
-Induced-Packing, -Scattered-Packing, and -Odd-Minor-Packing problem:
if is bipartite, then the problem is para-NP-complete parameterized by
bipartite treewidth while, if is non-bipartite, then it is solvable in
XP-time. We define 1--treewidth by replacing the bipartite graph
class by any class . Most of the technology developed here works for
this more general parameter.Comment: Presented in IPEC 202
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