89 research outputs found
On chordal phylogeny graphs
An acyclic digraph each vertex of which has indegree at most and
outdegree at most is called an digraph for some positive integers
and . Lee {\it et al.} (2017) studied the phylogeny graphs of
digraphs and gave sufficient conditions and necessary conditions for
digraphs having chordal phylogeny graphs. Their work was motivated by problems
related to evidence propagation in a Bayesian network for which it is useful to
know which acyclic digraphs have their moral graphs being chordal (phylogeny
graphs are called moral graphs in Bayesian network theory).
In this paper, we extend their work. We completely characterize phylogeny
graphs of digraphs and digraphs, respectively, for a positive
integer . Then, we study phylogeny graphs of a digraphs, which is
worthwhile in the context that a child has two biological parents in most
species, to show that the phylogeny graph of a digraph is chordal
if the underlying graph of is chordal for any positive integer .
Especially, we show that as long as the underlying graph of a digraph
is chordal, its phylogeny graph is not only chordal but also planar.Comment: 18 page
On weak majority dimensions of digraphs
In this paper, we introduce the notion of the weak majority dimension of a
digraph which is well-defined for any digraph. We first study properties shared
by the weak dimension of a digraph and show that a weak majority dimension of a
digraph can be arbitrarily large. Then we present a complete characterization
of digraphs of weak majority dimension and , respectively, and show that
every digraph with weak majority dimension at most two is transitive. Finally,
we compute the weak majority dimensions of directed paths and directed cycles
and pose open problems
The phylogeny number in the aspect of triangles and diamonds of a graph
Given an acyclic digraph , the competition graph of , denoted by
, is the simple graph having vertex set and edge set . The phylogeny graph
of an acyclic digraph , denoted by , is the graph with the vertex set
and the edge set where denotes the
underlying graph of . The notion of phylogeny graphs was introduced by
Roberts and Sheng~\cite{roberts1997phylogeny} as a variant of competition
graph. Moral graphs having arisen from studying Bayesian networks are the same
as phylogeny graphs. In this paper, we integrate the existing theorems
computing phylogeny numbers of connected graph with a small number of triangles
into one proposition: for a graph containing at most two triangle, where and
denote the number of triangles and the number of diamond in ,
respectively. Then we show that these inequalities hold for graphs with many
triangles. In the process of showing it, we derive a useful theorem which plays
a key role in deducing various meaningful results including a theorem that
answers a question given by Wu~{\it et al.}~\cite{Wu2019}
A new minimal chordal completion
In this paper, we present a minimal chordal completion of a graph
satisfying the inequality for the
non-chordality index of . In terms of our chordal completions, we
partially settle the Hadwiger conjecture and the Erd\H{o}s-Faber-Lov\'{a}sz
Conjecture, and extend the known -bounded class by adding to it the
family of graphs with bounded non-chordality indices
Competition numbers of planar graphs
In this paper, we relate the competition number of a graph to its edge clique
cover number by presenting a tight inequality where , , and
are the edge clique cover number, the competition number,
and the co-competition number of a graph , respectively. By utilizing this
inequality and a notion of competition-effective edge clique cover, we obtain
some meaningful results on competition numbers of planar graphs.Comment: 15 pages, 3 figure
On the safe set of Cartesian product of two complete graphs
For a connected graph , a vertex subset of is a safe set if for
every component of the subgraph of induced by , holds
for every component of such that there exists an edge between and
, and, in particular, if the subgraph induced by is connected, then
is called a connected safe set. For a connected graph , the safe number and
the connected safe number of are the minimum among sizes of the safe sets
and the minimum among sizes of the connected safe sets, respectively, of .
Fujita et al. introduced these notions in connection with a variation of the
facility location problem.
In this paper, we study the safe number and the connected safe number of
Cartesian product of two complete graphs. Figuring out a way to reduce the
number of components to two without changing the size of safe set makes it
sufficient to consider only partitions of an integer into two parts without
which it would be much more complicated to take care of all the partitions. In
this way, we could show that the safe number and the connected safe number of
Cartesian product of two complete graphs are equal and present a
polynomial-time algorithm to compute them. Especially, in the case where one of
complete components has order at most four, we precisely formulate those
numbers.Comment: 15 pages, 8 figure
On -step competition graphs of bipartite tournaments
In this paper, we completely characterize the -step competition graph of a
bipartite tournament for any integer . In addition, we compute the
competition index and the competition period of a bipartite tournament
Holes and a chordal cut in a graph
A set of vertices of a graph is called a {\em clique cut} of if
the subgraph of induced by is a complete graph and the number of
connected components of is greater than that of . A clique cut of
is called a {\em chordal cut} of if there exists a union of
connected components of such that is a chordal graph.
In this paper, we consider the following problem: Given a graph , does the
graph have a chordal cut? We show that -free hole-edge-disjoint
graphs have chordal cuts if they satisfy a certain condition.Comment: 12 pages, 1 figur
The competition number of a graph in which any two holes share at most one edge
The competition graph of a digraph D is a (simple undirected) graph which has
the same vertex set as D and has an edge between x and y if and only if there
exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph
G, G together with sufficiently many isolated vertices is the competition graph
of some acyclic digraph. The competition number k(G) of G is the smallest
number of such isolated vertices. In general, it is hard to compute the
competition number k(G) for a graph G and it has been one of important research
problems in the study of competition graphs to characterize a graph by its
competition number. A hole of a graph is a cycle of length at least 4 as an
induced subgraph. It holds that the competition number of a graph cannot exceed
one plus the number of its holes if G satisfies a certain condition. In this
paper, we show that the competition number of a graph with exactly h holes any
two of which share at most one edge is at most h+1, which generalizes the
existing results on this subject.Comment: 29 pages, 14 figures, 1 tabl
On consecutive edge magic total labeling of connected bipartite graphs
Since Sedl\'{a}\breve{\mbox{c}}ek introduced the notion of magic labeling
of a graph in 1963, a variety of magic labelings of a graph have been defined
and studied. In this paper, we study consecutive edge magic labelings of a
connected bipartite graph. We make a very useful observation that there are
only four possible values of for which a connected bipartite graph has a
-edge consecutive magic labeling. On the basis of this fundamental result,
we deduce various interesting results on consecutive edge magic labelings of
bipartite graphs, especially caterpillars and lobsters, which extends the
results given by Sugeng and Miller.Comment: 14 pages, 2 figure
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