19,073 research outputs found
Triangle-free -step competition graphs
Given a digraph and a positive number , the -step competition graph
of is defined to have the same vertex set as and an edge joining two
distinct vertices and if and only if there exist a -directed
walk and a -directed walk both having length for some vertex in
. We call a graph an -step competition graph if it is the -step
competition graph of a digraph. The notion of -step competition graph was
introduced by Cho \emph{et al.} \cite{cho2000m} as one of the variants of
competition graph which was introduced by Cohen \cite{Cohen} while studying
predator-prey concepts in ecological food webs.
In this paper, we first extend a result given by Helleloid
\cite{helleloid2005connected} stating that for all positive integers , the only connected triangle-free -step competition graph on vertices
is the star graph. We show that the result is true for arbitrary positive
integer as long as it is the -step competition graph of a digraph
having a source. We go further to completely characterize the digraphs each of
whose weak components has a source and whose -step competition graphs are
triangle-free for some integer . We also compute the number of
digraphs with a source whose -step competition graphs are connected and
triangle-free for some integer
์ํ๊ณ์์์ ๊ฒฝ์ ๊ด์ ์ผ๋ก ๊ทธ๋ํ์ ์ ํฅ๊ทธ๋ํ์ ๊ตฌ์กฐ ์ฐ๊ตฌ
ํ์๋
ผ๋ฌธ(๋ฐ์ฌ) -- ์์ธ๋ํ๊ต๋ํ์ : ์ฌ๋ฒ๋ํ ์ํ๊ต์ก๊ณผ, 2023. 2. ๊น์๋ น.In this thesis, we study m-step competition graphs, (1, 2)-step competition graphs, phylogeny graphs, and competition-common enemy graphs (CCE graphs), which are primary variants of competition graphs. Cohen [11] introduced the notion of competition graph while studying predator-prey concepts in ecological food webs.An ecosystem is a biological community of interacting species and their physical environment. For each species in an ecosystem, there can be m conditions of the good environment by regarding lower and upper bounds on numerous dimensions such as soil, climate, temperature, etc, which may be represented by an m-dimensional rectangle, so-called an ecological niche. An elemental ecological truth is that two species compete if and only if their ecological niches overlap. Biologists often describe competitive relations among species cohabiting in a community by a food web that is a digraph whose vertices are the species and an arc goes from a predator to a prey. In this context, Cohen [11] defined the competition graph of a digraph as follows. The competition graph C(D) of a digraph D is defined to be a simple graph whose vertex set is the same as V (D) and which has an edge joining two distinct vertices u and v if and only if there are arcs (u, w) and (v, w) for some vertex w in D. Since Cohen introduced this definition, its variants such as m-step competition graphs, (i, j)-step competition graphs, phylogeny graphs, CCE graphs, p-competition graphs, and niche graphs have been introduced and studied.
As part of these studies, we show that the connected triangle-free m-step competition graph on n vertices is a tree and completely characterize the digraphs of order n whose m-step competition graphs are star graphs for positive integers 2 โค m < n.
We completely identify (1,2)-step competition graphs C_{1,2}(D) of orientations D of a complete k-partite graph for some k โฅ 3 when each partite set of D forms a clique in C_{1,2}(D). In addition, we show that the diameter of each component of C_{1,2}(D) is at most three and provide a sharp upper bound on the domination number of C_{1,2}(D) and give a sufficient condition for C_{1,2}(D) being an interval graph.
On the other hand, we study on phylogeny graphs and CCE graphs of degreebounded acyclic digraphs. An acyclic digraph in which every vertex has indegree at most i and outdegree at most j is called an (i, j) digraph for some positive integers i and j. If each vertex of a (not necessarily acyclic) digraph D has indegree at most i and outdegree at most j, then D is called an hi, ji digraph. We give a sufficient condition on the size of hole of an underlying graph of an (i, 2) digraph D for the phylogeny graph of D being a chordal graph where D is an (i, 2) digraph. Moreover, we go further to completely characterize phylogeny graphs of (i, j) digraphs by listing the forbidden induced subgraphs.
We completely identify the graphs with the least components among the CCE graphs of (2, 2) digraphs containing at most one cycle and exactly two isolated vertices, and their digraphs. Finally, we gives a sufficient condition for CCE graphs
being interval graphs.์ด ๋
ผ๋ฌธ์์ ๊ฒฝ์๊ทธ๋ํ์ ์ฃผ์ ๋ณ์ด๋ค ์ค m-step ๊ฒฝ์๊ทธ๋ํ, (1, 2)-step ๊ฒฝ์ ๊ทธ๋ํ, ๊ณํต ๊ทธ๋ํ, ๊ฒฝ์๊ณต์ ๊ทธ๋ํ์ ๋ํ ์ฐ๊ตฌ ๊ฒฐ๊ณผ๋ฅผ ์ข
ํฉํ๋ค. Cohen [11]์ ๋จน์ด์ฌ์ฌ์์ ํฌ์์-ํผ์์ ๊ฐ๋
์ ์ฐ๊ตฌํ๋ฉด์ ๊ฒฝ์๊ทธ๋ํ ๊ฐ๋
์ ๊ณ ์ํ๋ค. ์ํ๊ณ๋ ์ํธ์์ฉํ๋ ์ข
๋ค๊ณผ ๊ทธ๋ค์ ๋ฌผ๋ฆฌ์ ํ๊ฒฝ์ ์๋ฌผํ์ ์ฒด๊ณ์ด๋ค. ์ํ๊ณ์ ๊ฐ ์ข
์ ๋ํด์, ํ ์, ๊ธฐํ, ์จ๋ ๋ฑ๊ณผ ๊ฐ์ ๋ค์ํ ์ฐจ์์ ํ๊ณ ๋ฐ ์๊ณ๋ฅผ ๊ณ ๋ คํ์ฌ ์ข์ ํ๊ฒฝ์ m๊ฐ์ ์กฐ๊ฑด๋ค๋ก ๋ํ๋ผ ์ ์๋๋ฐ ์ด๋ฅผ ์ํ์ ์ง์(ecological niche)๋ผ๊ณ ํ๋ค. ์ํํ์ ๊ธฐ๋ณธ๊ฐ์ ์ ๋ ์ข
์ด ์ํ์ ์ง์๊ฐ ๊ฒน์น๋ฉด ๊ฒฝ์ํ๊ณ (compete), ๊ฒฝ์ํ๋ ๋ ์ข
์ ์ํ์ ์ง์๊ฐ ๊ฒน์น๋ค๋ ๊ฒ์ด๋ค. ํํ ์๋ฌผํ์๋ค์ ํ ์ฒด์ ์์ ์์ํ๋ ์ข
๋ค์ ๊ฒฝ์์ ๊ด๊ณ๋ฅผ ๊ฐ ์ข
์ ๊ผญ์ง์ ์ผ๋ก, ํฌ์์์์ ํผ์์์๊ฒ๋ ์ ํฅ๋ณ(arc)์ ๊ทธ์ด์ ๋จน์ด์ฌ์ฌ๋ก ํํํ๋ค. ์ด๋ฌํ ๋งฅ๋ฝ์์ Cohen [11]์ ๋ค์๊ณผ ๊ฐ์ด ์ ํฅ๊ทธ๋ํ์ ๊ฒฝ์ ๊ทธ๋ํ๋ฅผ ์ ์ํ๋ค. ์ ํฅ๊ทธ๋ํ(digraph) D์ ๊ฒฝ์๊ทธ๋ํ(competition graph) C(D) ๋ V (D)๋ฅผ ๊ผญ์ง์ ์งํฉ์ผ๋ก ํ๊ณ ๋ ๊ผญ์ง์ u, v๋ฅผ ์ ๋์ ์ผ๋ก ๊ฐ๋ ๋ณ์ด ์กด์ฌํ๋ค๋ ๊ฒ๊ณผ ๊ผญ์ง์ w๊ฐ ์กด์ฌํ์ฌ (u, w),(v, w)๊ฐ ๋ชจ๋ D์์ ์ ํฅ๋ณ์ด ๋๋ ๊ฒ์ด ๋์น์ธ ๊ทธ๋ํ๋ฅผ ์๋ฏธํ๋ค. Cohen์ด ๊ฒฝ์๊ทธ๋ํ์ ์ ์๋ฅผ ๋์
ํ ์ดํ๋ก ๊ทธ ๋ณ์ด๋ค๋ก m-step ๊ฒฝ์๊ทธ๋ํ(m-step competition graph), (i, j)-step ๊ฒฝ์๊ทธ๋ํ((i, j)-step competition graph), ๊ณํต๊ทธ๋ํ(phylogeny graph), ๊ฒฝ์๊ณต์ ๊ทธ๋ํ(competition-common enemy graph), p-๊ฒฝ์๊ทธ๋ํ(p-competition graph), ๊ทธ๋ฆฌ๊ณ ์ง์๊ทธ๋ํ(niche graph)๊ฐ ๋์
๋์๊ณ ์ฐ๊ตฌ๋๊ณ ์๋ค.
์ด ๋
ผ๋ฌธ์ ์ฐ๊ตฌ ๊ฒฐ๊ณผ๋ค์ ์ผ๋ถ๋ ๋ค์๊ณผ ๊ฐ๋ค. ์ผ๊ฐํ์ด ์์ด ์ฐ๊ฒฐ๋ m-step ๊ฒฝ์ ๊ทธ๋ํ๋ ํธ๋ฆฌ(tree)์์ ๋ณด์์ผ๋ฉฐ 2 โค m < n์ ๋ง์กฑํ๋ ์ ์ m, n์ ๋ํ์ฌ ๊ผญ์ง์ ์ ๊ฐ์๊ฐ n๊ฐ์ด๊ณ m-step ๊ฒฝ์๊ทธ๋ํ๊ฐ ๋ณ๊ทธ๋ํ(star graph)๊ฐ ๋๋ ์ ํฅ๊ทธ๋ํ๋ฅผ ์๋ฒฝํ๊ฒ ํน์งํ ํ์๋ค.
k โฅ 3์ด๊ณ ๋ฐฉํฅ์ง์ด์ง ์์ k-๋ถํ ๊ทธ๋ํ(oriented complete k-partite graph)์ (1, 2)-step ๊ฒฝ์๊ทธ๋ํ C_{1,2}(D)์์ ๊ฐ ๋ถํ ์ด ์์ ๋ถ๋ถ ๊ทธ๋ํ๋ฅผ ์ด๋ฃฐ ๋, C_{1,2}(D)์ ๋ชจ๋ ํน์งํ ํ์๋ค. ๋ํ, C_{1,2}(D)์ ๊ฐ ์ฑ๋ถ(component)์ ์ง๋ฆ(diameter)์ ๊ธธ์ด๊ฐ ์ต๋ 3์ด๋ฉฐ C_{1,2}(D)์ ์ง๋ฐฐ์(domination number)์ ๋ํ ์๊ณ์ ์ต๋๊ฐ์ ๊ตฌํ๊ณ ๊ตฌ๊ฐ๊ทธ๋ํ(interval graph)๊ฐ ๋๊ธฐ ์ํ ์ถฉ๋ถ ์กฐ๊ฑด์ ๊ตฌํ์๋ค.
์ฐจ์๊ฐ ์ ํ๋ ์ ํฅํ๋ก๋ฅผ ๊ฐ์ง ์๋ ์ ํฅ๊ทธ๋ํ(degree-bounded acyclic digraph)์ ๊ณํต๊ทธ๋ํ์ ๊ฒฝ์๊ณต์ ๊ทธ๋ํ์ ๋ํด์๋ ์ฐ๊ตฌํ์๋ค. ์์ ์ ์๋ค i, j์ ๋ํ์ฌ (i, j) ์ ํฅ๊ทธ๋ํ๋ ๊ฐ ๊ผญ์ง์ ์ ๋ด์ฐจ์๋ ์ต๋ i, ์ธ์ฐจ์๋ ์ต๋ j์ธ ์ ํฅํ๋ก ๊ฐ์ง ์๋ ์ ํฅ๊ทธ๋ํ์ด๋ค. ๋ง์ฝ ์ ํฅ๊ทธ๋ํ D์ ๊ฐ ๊ผญ์ง์ ์ด ๋ด์ฐจ์๊ฐ ์ต๋ i, ์ธ์ฐจ์๊ฐ ์ต๋ j ์ธ ๊ฒฝ์ฐ์ D๋ฅผ hi, ji ์ ํฅ๊ทธ๋ํ๋ผ ํ๋ค.
D๊ฐ (i, 2) ์ ํฅ๊ทธ๋ํ์ผ ๋, D์ ๊ณํต๊ทธ๋ํ๊ฐ ํ๊ทธ๋ํ(chordal graph)๊ฐ ๋๊ธฐ ์ํ D์ ๋ฐฉํฅ์ ๊ณ ๋ คํ์ง ์๊ณ ์ป์ด์ง๋ ๊ทธ๋ํ(underlying graph)์์ ๊ธธ์ด๊ฐ 4์ด์์ธ ํ๋ก(hole)์ ๊ธธ์ด์ ๋ํ ์ถฉ๋ถ์กฐ๊ฑด์ ๊ตฌํ์๋ค. ๊ฒ๋ค๊ฐ (i, j) ์ ํฅ๊ทธ๋ํ์ ๊ณํต๊ทธ๋ํ์์ ๋์ฌ ์ ์๋ ์์ฑ ๋ถ๋ถ ๊ทธ๋ํ(forbidden induced subgraph)๋ฅผ ํน์งํ ํ์๋ค.
(2, 2) ์ ํฅ๊ทธ๋ํ D์ ๊ฒฝ์๊ณต์ ๊ทธ๋ํ CCE(D)๊ฐ 2๊ฐ์ ๊ณ ๋ฆฝ์ (isolated vertex)๊ณผ ์ต๋ 1๊ฐ์ ํ๋ก๋ฅผ ๊ฐ์ผ๋ฉด์ ๊ฐ์ฅ ์ ์ ์ฑ๋ถ์ ๊ฐ๋ ๊ฒฝ์ฐ์ผ ๋์ ๊ตฌ์กฐ๋ฅผ ๊ท๋ช
ํ๋ค. ๋ง์ง๋ง์ผ๋ก, CCE(D)๊ฐ ๊ตฌ๊ฐ๊ทธ๋ํ๊ฐ ๋๊ธฐ ์ํ ์ฑ๋ถ์ ๊ฐ์์ ๋ํ ์ถฉ๋ถ์กฐ๊ฑด์ ๊ตฌํ์๋ค.1 Introduction 1
1.1 Graph theory terminology and basic concepts 1
1.2 Competition graphs and its variants 6
1.2.1 A brief background of competition graphs 6
1.2.2 Variants of competition graphs 8
1.2.3 m-step competition graphs 10
1.2.4 (1, 2)-step competition graphs 13
1.2.5 Phylogeny graphs 14
1.2.6 CCE graphs 16
1.3 A preview of the thesis 17
2 Digraphs whose m-step competition graphs are trees 19
2.1 The triangle-free m-step competition graphs 23
2.2 Digraphs whose m-step competition graphs are trees 29
2.3 The digraphs whose m-step competition graphs are star graphs 38
3 On (1, 2)-step competition graphs of multipartite tournaments 47
3.1 Preliminaries 48
3.2 C1,2(D) with a non-clique partite set of D 51
3.3 C1,2(D) without a non-clique partite set of D 66
3.4 C1,2(D) as a complete graph 74
3.5 Diameters and domination numbers of C1,2(D) 79
3.6 Disconnected (1, 2)-step competition graphs 82
3.7 Interval (1, 2)-step competition graphs 84
4 The forbidden induced subgraphs of (i, j) phylogeny graphs 90
4.1 A necessary condition for an (i, 2) phylogeny graph being chordal 91
4.2 Forbidden subgraphs for phylogeny graphs of degree bounded digraphs 99
5 On CCE graphs of (2, 2) digraphs 122
5.1 CCE graphs of h2, 2i digraphs 128
5.2 CCE graphs of (2, 2) digraphs 134
Abstract (in Korean) 168
Acknowledgement (in Korean) 170๋ฐ
The competition numbers of ternary Hamming graphs
It is known to be a hard problem to compute the competition number k(G) of a
graph G in general. Park and Sano [13] gave the exact values of the competition
numbers of Hamming graphs H(n,q) if or . In
this paper, we give an explicit formula of the competition numbers of ternary
Hamming graphs.Comment: 6 pages, 2 figure
Coexistence of competing first passage percolation on hyperbolic graphs
We study a natural growth process with competition, which was recently
introduced to analyze MDLA, a challenging model for the growth of an aggregate
by diffusing particles. The growth process consists of two first-passage
percolation processes and , spreading with
rates and respectively, on a graph . starts
from a single vertex at the origin , while the initial configuration of
consists of infinitely many \emph{seeds} distributed
according to a product of Bernoulli measures of parameter on
. starts spreading from time 0, while each
seed of only starts spreading after it has been reached by
either or . A fundamental question in this
model, and in growth processes with competition in general, is whether the two
processes coexist (i.e., both produce infinite clusters) with positive
probability. We show that this is the case when is vertex transitive,
non-amenable and hyperbolic, in particular, for any there is a
such that for all the two
processes coexist with positive probability. This is the first non-trivial
instance where coexistence is established for this model. We also show that
produces an infinite cluster almost surely for any
positive , establishing fundamental differences with the behavior
of such processes on .Comment: 53 pages, 13 figure
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