19 research outputs found
A linear algorithm for the grundy number of a tree
A coloring of a graph G = (V,E) is a partition {V1, V2, . . ., Vk} of V into
independent sets or color classes. A vertex v Vi is a Grundy vertex if it is
adjacent to at least one vertex in each color class Vj . A coloring is a Grundy
coloring if every color class contains at least one Grundy vertex, and the
Grundy number of a graph is the maximum number of colors in a Grundy coloring.
We derive a natural upper bound on this parameter and show that graphs with
sufficiently large girth achieve equality in the bound. In particular, this
gives a linear time algorithm to determine the Grundy number of a tree
Grundy Coloring & Friends, Half-Graphs, Bicliques
The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order ?, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering ?, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f(k)n^{2^{k-1}}-time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative.
The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS \u2717]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on K_{t,t}-free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest
Complexity of Grundy coloring and its variants
The Grundy number of a graph is the maximum number of colors used by the
greedy coloring algorithm over all vertex orderings. In this paper, we study
the computational complexity of GRUNDY COLORING, the problem of determining
whether a given graph has Grundy number at least . We also study the
variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper)
and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring
algorithm, the subgraph induced by the colored vertices must be connected).
We show that GRUNDY COLORING can be solved in time and WEAK
GRUNDY COLORING in time on graphs of order . While GRUNDY
COLORING and WEAK GRUNDY COLORING are known to be solvable in time
for graphs of treewidth (where is the number of
colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot
be solved in time . We also describe an
algorithm for WEAK GRUNDY COLORING, which is therefore
\fpt for the parameter . Moreover, under the ETH, we prove that such a
running time is essentially optimal (this lower bound also holds for GRUNDY
COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we
show that this is the case for graphs belonging to a number of standard graph
classes including chordal graphs, claw-free graphs, and graphs excluding a
fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY
COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with
the two other problems, we show that CONNECTED GRUNDY COLORING is
\np-complete already for colors.Comment: 24 pages, 7 figures. This version contains some new results and
improvements. A short paper based on version v2 appeared in COCOON'1
Transitivity on subclasses of chordal graphs
Let be a graph, where and are the vertex and edge sets,
respectively. For two disjoint subsets and of , we say
\textit{dominates} if every vertex of is adjacent to at least one
vertex of in . A vertex partition of
is called a \emph{transitive -partition} if dominates for
all , where . The maximum integer for which the above
partition exists is called \emph{transitivity} of and it is denoted by
. The \textsc{Maximum Transitivity Problem} is to find a transitive
partition of a given graph with the maximum number of partitions. It was known
that the decision version of \textsc{Maximum Transitivity Problem} is
NP-complete for chordal graphs [Iterated colorings of graphs, \emph{Discrete
Mathematics}, 278, 2004]. In this paper, we first prove that this problem can
be solved in linear time for \emph{split graphs} and for the \emph{complement
of bipartite chain graphs}, two subclasses of chordal graphs. We also discuss
Nordhaus-Gaddum type relations for transitivity and provide counterexamples for
an open problem posed by J. T. Hedetniemi and S. T. Hedetniemi [The
transitivity of a graph, \emph{J. Combin. Math. Combin. Comput}, 104, 2018].
Finally, we characterize transitively critical graphs having fixed
transitivity.Comment: arXiv admin note: text overlap with arXiv:2204.1314
Entropy and Graphs
The entropy of a graph is a functional depending both on the graph itself and
on a probability distribution on its vertex set. This graph functional
originated from the problem of source coding in information theory and was
introduced by J. K\"{o}rner in 1973. Although the notion of graph entropy has
its roots in information theory, it was proved to be closely related to some
classical and frequently studied graph theoretic concepts. For example, it
provides an equivalent definition for a graph to be perfect and it can also be
applied to obtain lower bounds in graph covering problems.
In this thesis, we review and investigate three equivalent definitions of
graph entropy and its basic properties. Minimum entropy colouring of a graph
was proposed by N. Alon in 1996. We study minimum entropy colouring and its
relation to graph entropy. We also discuss the relationship between the entropy
and the fractional chromatic number of a graph which was already established in
the literature.
A graph is called \emph{symmetric with respect to a functional }
defined on the set of all the probability distributions on its vertex set if
the distribution maximizing is uniform on . Using the
combinatorial definition of the entropy of a graph in terms of its vertex
packing polytope and the relationship between the graph entropy and fractional
chromatic number, we prove that vertex transitive graphs are symmetric with
respect to graph entropy. Furthermore, we show that a bipartite graph is
symmetric with respect to graph entropy if and only if it has a perfect
matching. As a generalization of this result, we characterize some classes of
symmetric perfect graphs with respect to graph entropy. Finally, we prove that
the line graph of every bridgeless cubic graph is symmetric with respect to
graph entropy.Comment: 89 pages, 4 figures, MMath Thesi
A Study Of The Upper Domatic Number Of A Graph
Given a graph G we can partition the vertices of G in to k disjoint sets. We say a set A of vertices dominates another set of vertices, B, if for every vertex in B there is some adjacent vertex in A. The upper domatic number of a graph G is written as D(G) and defined as the maximum integer k such that G can be partitioned into k sets where for every pair of sets A and B either A dominates B or B dominates A or both. In this thesis we introduce the upper domatic number of a graph and provide various results on the properties of the upper domatic number, notably that D(G) is less than or equal to the maximum degree of G, as well as relating it to other well-studied graph properties such as the achromatic, pseudoachromatic, and transitive numbers