138 research outputs found
A forbidden subgraph characterization of phylogeny graphs of degree bounded digraphs
An acyclic digraph in which every vertex has indegree at most and
outdegree at most is called an digraph for some positive integers
and . The phylogeny graph of a digraph has as the vertex set
and an edge if and only if one of the following is true: ;
; and for some .
A graph is a phylogeny graph (resp. an phylogeny graph) if there is
an acyclic digraph (resp. an digraph ) such that the phylogeny
graph of is isomorphic to . Lee~{\em et al.} (2017) and Eoh and Kim
(2021) studied the phylogeny graphs, phylogeny graphs,
phylogeny graphs, and 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 chordal moral graphs (phylogeny
graphs are called moral graphs in Bayesian network theory). In this paper, we
extend their work by characterizing chordal phylogeny graphs. We go
further to completely characterize phylogeny graphs by listing the
forbidden induced subgraphs
Competition graphs of degree bounded digraphs
If each vertex of an acyclic digraph has indegree at most and outdegree
at most , then it is called an digraph, which was introduced by
Hefner~{\it et al.}~(1991). Whereas Hefner~{\it et al.} characterized
digraphs whose competition graphs are interval, characterizing the competition
graphs of digraphs is not an easy task. In this paper, we introduce the
concept of digraphs, which relax the acyclicity condition
of digraphs, and study their competition graphs. By doing so, we obtain
quite meaningful results. Firstly, we give a necessary and sufficient condition
for a loopless graph being an competition graph for some
positive integers and . Then we study on an
competition graph being chordal and present a forbidden subdigraph
characterization. Finally, we study the family of
competition graphs, denoted by , and
identify the set containment relation on
The Complexity of Surjective Homomorphism Problems -- a Survey
We survey known results about the complexity of surjective homomorphism
problems, studied in the context of related problems in the literature such as
list homomorphism, retraction and compaction. In comparison with these
problems, surjective homomorphism problems seem to be harder to classify and we
examine especially three concrete problems that have arisen from the
literature, two of which remain of open complexity
07281 Abstracts Collection -- Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs
From 8th to 13th July 2007, the Dagstuhl Seminar ``Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Holeμ κ΄μ μμ κ·Έλνμ μ ν₯κ·Έλνμ ꡬ쑰μ κ΄ν μ°κ΅¬
νμλ
Όλ¬Έ(λ°μ¬)--μμΈλνκ΅ λνμ :μ¬λ²λν μνκ΅μ‘κ³Ό,2019. 8. κΉμλ Ή.μ΄ λ
Όλ¬Έμμλ μ ν₯κ·Έλνμ κ·Έλνμ νμ κ΄μ μμ κ³ν΅λ°μ κ·Έλνμ κ·Έλνμ μΌκ°νμ λνμ¬ μ°κ΅¬νλ€.
κΈΈμ΄ 4 μ΄μμΈ μ λλ μΈμ΄ν΄μ νμ΄λΌ νκ³ νμ΄ μλ κ·Έλνλ₯Ό μΌκ°νλ κ·ΈλνλΌ νλ€. ꡬ체μ μΌλ‘, μΈμ΄ν΄μ κ°μ§ μλ μ ν₯κ·Έλνμ κ³ν΅λ°μ κ·Έλνκ° μΌκ°νλ κ·ΈλνμΈμ§ νμ νκ³ , μ£Όμ΄μ§ κ·Έλνλ₯Ό μΌκ°ννμ¬ ν΄λ¦μκ° ν¬κ² μ°¨μ΄ λμ§ μλ κ·Έλνλ₯Ό λ§λλ λ°©λ²μ μ°Ύκ³ μ νλ€. μ΄ λ
Όλ¬Έμ μ°κ΅¬ λ΄μ©μ λ°λΌ λ λΆλΆμΌλ‘ λλλ€.
λ¨Όμ μ ν₯κ·Έλνμ μ ν₯κ·Έλνμ κ³ν΅λ°μ κ·Έλνλ₯Ό μμ νκ² νΉμ§ννκ³ , μ ν₯κ·Έλν μ λͺ¨λ μ ν₯λ³μμ λ°©ν₯μ μ κ±°ν κ·Έλνκ° μΌκ°νλ κ·Έλνμ΄λ©΄, μ κ³ν΅λ°μ κ·Έλν μμ μΌκ°νλ κ·Έλνμμ 보μλ€. λν μ μ μμ μΌκ°νμ κ°λ μ°κ²°λ κ·Έλνμ κ³ν΅λ°μμλ₯Ό κ³μ°ν μ 리λ₯Ό νμ₯νμ¬ λ§μ μμ μΌκ°νμ ν¬ν¨ν μ°κ²°λ κ·Έλνμ κ³ν΅λ°μμλ₯Ό κ³μ°νμλ€.
λ€λ₯Έ ν νΈ κ·Έλν μ λΉμΌκ°ν μ§μ μ λνμ¬ λ₯Ό λ§μ‘±νλ μ μΌκ°νλ κ·Έλν κ° μ‘΄μ¬ν¨μ 보μλ€.
κ·Έλ¦¬κ³ μ΄λ₯Ό λκ΅¬λ‘ μ΄μ©νμ¬ NC propertyλ₯Ό λ§μ‘±νλ κ·Έλνκ° Hadwiger μΆμΈ‘κ³Ό Erd\H{o}s-Faber-Lov\'{a}sz μΆμΈ‘μ λ§μ‘±ν¨μ μ¦λͺ
νκ³ , λΉμΌκ°ν μ§μκ° μ κ³μΈ κ·Έλνλ€μ΄ linearly -boundedμμ μ¦λͺ
νμλ€.This thesis aims at studying phylogeny graphs and graph completions in the aspect of holes of graphs or digraphs. A hole of a graph is an induced cycle of length at least four and a graph is chordal if it does not contain a hole. Specifically, we determine whether the phylogeny graphs of acyclic digraphs are chordal or not and find a way of chordalizing a graph without increasing the size of maximum clique not so much. In this vein, the thesis is divided into two parts.
In the first part, we completely characterize phylogeny graphs of digraphs and digraphs, respectively, for a positive integer . Then, we show that the phylogeny graph of a digraph is chordal if the underlying graph of is chordal for any positive integer . In addition, we extend the existing theorems computing phylogeny numbers of connected graph with a small number of triangles to results computing phylogeny numbers of connected graphs with many triangles.
In the second part, we present a minimal chordal supergraph of a graph satisfying the inequality for the non-chordality index of . Using the above chordal supergraph as a tool, we prove that the family of graphs satisfying the NC property satisfies the Hadwiger conjecture and the Erd\H{o}s-Faber-Lov\'{a}sz Conjecture, and the family of graphs with bounded non-chordality indices is linearly -bounded.Contents
Abstract i
1 Introduction 1
1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Phylogeny graphs . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Graph colorings and chordal completions . . . . . . . . 14
2 Phylogeny graphs 19
2.1 Chordal phylogeny graphs . . . . . . . . . . . . . . . . . . . . 19
2.1.1 (1,j) phylogeny graphs and (i,1) phylogeny graphs . . 20
2.1.2 (2,j) phylogeny graphs . . . . . . . . . . . . . . . . . . 28
2.2 The phylogeny number and the triangles and the diamonds of
a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 A new minimal chordal completion 61
3.1 Graphs with the NC property . . . . . . . . . . . . . . . . . . 64
3.2 The ErdΛ os-Faber-LovΓ‘sz Conjecture . . . . . . . . . . . . . . . 73
3.3 A minimal chordal completion of a graph . . . . . . . . . . . . 80
3.3.1 Non-chordality indices of graphs . . . . . . . . . . . . . 80
3.3.2 Making a local chordalization really local . . . . . . . . 89
3.4 New Ο-bounded classes . . . . . . . . . . . . . . . . . . . . . . 97
Abstract (in Korean) 107Docto
Reconstructing Gene Trees From Fitch's Xenology Relation
Two genes are xenologs in the sense of Fitch if they are separated by at
least one horizontal gene transfer event. Horizonal gene transfer is asymmetric
in the sense that the transferred copy is distinguished from the one that
remains within the ancestral lineage. Hence xenology is more precisely thought
of as a non-symmetric relation: is xenologous to if has been
horizontally transferred at least once since it diverged from the least common
ancestor of and . We show that xenology relations are characterized by a
small set of forbidden induced subgraphs on three vertices. Furthermore, each
xenology relation can be derived from a unique least-resolved edge-labeled
phylogenetic tree. We provide a linear-time algorithm for the recognition of
xenology relations and for the construction of its least-resolved edge-labeled
phylogenetic tree. The fact that being a xenology relation is a heritable graph
property, finally has far-reaching consequences on approximation problems
associated with xenology relations
μνκ³μμμ κ²½μ κ΄μ μΌλ‘ κ·Έλνμ μ ν₯κ·Έλνμ ꡬ쑰 μ°κ΅¬
νμλ
Όλ¬Έ(λ°μ¬) -- μμΈλνκ΅λνμ : μ¬λ²λν μνκ΅μ‘κ³Ό, 2023. 2. κΉμλ Ή.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 β€ 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 β₯ 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.μ΄ λ
Όλ¬Έμμ κ²½μκ·Έλνμ μ£Όμ λ³μ΄λ€ μ€ m-step κ²½μκ·Έλν, (1, 2)-step κ²½μ κ·Έλν, κ³ν΅ κ·Έλν, κ²½μ곡μ κ·Έλνμ λν μ°κ΅¬ κ²°κ³Όλ₯Ό μ’
ν©νλ€. Cohen [11]μ λ¨Ήμ΄μ¬μ¬μμ ν¬μμ-νΌμμ κ°λ
μ μ°κ΅¬νλ©΄μ κ²½μκ·Έλν κ°λ
μ κ³ μνλ€. μνκ³λ μνΈμμ©νλ μ’
λ€κ³Ό κ·Έλ€μ 물리μ νκ²½μ μλ¬Όνμ 체κ³μ΄λ€. μνκ³μ κ° μ’
μ λν΄μ, ν μ, κΈ°ν, μ¨λ λ±κ³Ό κ°μ λ€μν μ°¨μμ νκ³ λ° μκ³λ₯Ό κ³ λ €νμ¬ μ’μ νκ²½μ mκ°μ 쑰건λ€λ‘ λνλΌ μ μλλ° μ΄λ₯Ό μνμ μ§μ(ecological niche)λΌκ³ νλ€. μννμ κΈ°λ³Έκ°μ μ λ μ’
μ΄ μνμ μ§μκ° κ²ΉμΉλ©΄ κ²½μνκ³ (compete), κ²½μνλ λ μ’
μ μνμ μ§μκ° κ²ΉμΉλ€λ κ²μ΄λ€. νν μλ¬Όνμλ€μ ν 체μ μμ μμνλ μ’
λ€μ κ²½μμ κ΄κ³λ₯Ό κ° μ’
μ κΌμ§μ μΌλ‘, ν¬μμμμ νΌμμμκ²λ μ ν₯λ³(arc)μ κ·Έμ΄μ λ¨Ήμ΄μ¬μ¬λ‘ νννλ€. μ΄λ¬ν λ§₯λ½μμ Cohen [11]μ λ€μκ³Ό κ°μ΄ μ ν₯κ·Έλνμ κ²½μ κ·Έλνλ₯Ό μ μνλ€. μ ν₯κ·Έλν(digraph) Dμ κ²½μκ·Έλν(competition graph) C(D) λ V (D)λ₯Ό κΌμ§μ μ§ν©μΌλ‘ νκ³ λ κΌμ§μ u, vλ₯Ό μ λμ μΌλ‘ κ°λ λ³μ΄ μ‘΄μ¬νλ€λ κ²κ³Ό κΌμ§μ wκ° μ‘΄μ¬νμ¬ (u, w),(v, w)κ° λͺ¨λ Dμμ μ ν₯λ³μ΄ λλ κ²μ΄ λμΉμΈ κ·Έλνλ₯Ό μλ―Ένλ€. Cohenμ΄ κ²½μκ·Έλνμ μ μλ₯Ό λμ
ν μ΄νλ‘ κ·Έ λ³μ΄λ€λ‘ m-step κ²½μκ·Έλν(m-step competition graph), (i, j)-step κ²½μκ·Έλν((i, j)-step competition graph), κ³ν΅κ·Έλν(phylogeny graph), κ²½μ곡μ κ·Έλν(competition-common enemy graph), p-κ²½μκ·Έλν(p-competition graph), κ·Έλ¦¬κ³ μ§μκ·Έλν(niche graph)κ° λμ
λμκ³ μ°κ΅¬λκ³ μλ€.
μ΄ λ
Όλ¬Έμ μ°κ΅¬ κ²°κ³Όλ€μ μΌλΆλ λ€μκ³Ό κ°λ€. μΌκ°νμ΄ μμ΄ μ°κ²°λ m-step κ²½μ κ·Έλνλ νΈλ¦¬(tree)μμ 보μμΌλ©° 2 β€ m < nμ λ§μ‘±νλ μ μ m, nμ λνμ¬ κΌμ§μ μ κ°μκ° nκ°μ΄κ³ m-step κ²½μκ·Έλνκ° λ³κ·Έλν(star graph)κ° λλ μ ν₯κ·Έλνλ₯Ό μλ²½νκ² νΉμ§ν νμλ€.
k β₯ 3μ΄κ³ λ°©ν₯μ§μ΄μ§ μμ k-λΆν κ·Έλν(oriented complete k-partite graph)μ (1, 2)-step κ²½μκ·Έλν C_{1,2}(D)μμ κ° λΆν μ΄ μμ λΆλΆ κ·Έλνλ₯Ό μ΄λ£° λ, C_{1,2}(D)μ λͺ¨λ νΉμ§ν νμλ€. λν, C_{1,2}(D)μ κ° μ±λΆ(component)μ μ§λ¦(diameter)μ κΈΈμ΄κ° μ΅λ 3μ΄λ©° C_{1,2}(D)μ μ§λ°°μ(domination number)μ λν μκ³μ μ΅λκ°μ ꡬνκ³ κ΅¬κ°κ·Έλν(interval graph)κ° λκΈ° μν μΆ©λΆ μ‘°κ±΄μ ꡬνμλ€.
μ°¨μκ° μ νλ μ ν₯νλ‘λ₯Ό κ°μ§ μλ μ ν₯κ·Έλν(degree-bounded acyclic digraph)μ κ³ν΅κ·Έλνμ κ²½μ곡μ κ·Έλνμ λν΄μλ μ°κ΅¬νμλ€. μμ μ μλ€ i, jμ λνμ¬ (i, j) μ ν₯κ·Έλνλ κ° κΌμ§μ μ λ΄μ°¨μλ μ΅λ i, μΈμ°¨μλ μ΅λ jμΈ μ ν₯νλ‘ κ°μ§ μλ μ ν₯κ·Έλνμ΄λ€. λ§μ½ μ ν₯κ·Έλν Dμ κ° κΌμ§μ μ΄ λ΄μ°¨μκ° μ΅λ i, μΈμ°¨μκ° μ΅λ j μΈ κ²½μ°μ Dλ₯Ό hi, ji μ ν₯κ·ΈλνλΌ νλ€.
Dκ° (i, 2) μ ν₯κ·ΈλνμΌ λ, Dμ κ³ν΅κ·Έλνκ° νκ·Έλν(chordal graph)κ° λκΈ° μν Dμ λ°©ν₯μ κ³ λ €νμ§ μκ³ μ»μ΄μ§λ κ·Έλν(underlying graph)μμ κΈΈμ΄κ° 4μ΄μμΈ νλ‘(hole)μ κΈΈμ΄μ λν μΆ©λΆμ‘°κ±΄μ ꡬνμλ€. κ²λ€κ° (i, j) μ ν₯κ·Έλνμ κ³ν΅κ·Έλνμμ λμ¬ μ μλ μμ± λΆλΆ κ·Έλν(forbidden induced subgraph)λ₯Ό νΉμ§ν νμλ€.
(2, 2) μ ν₯κ·Έλν Dμ κ²½μ곡μ κ·Έλν CCE(D)κ° 2κ°μ κ³ λ¦½μ (isolated vertex)κ³Ό μ΅λ 1κ°μ νλ‘λ₯Ό κ°μΌλ©΄μ κ°μ₯ μ μ μ±λΆμ κ°λ κ²½μ°μΌ λμ ꡬ쑰λ₯Ό κ·λͺ
νλ€. λ§μ§λ§μΌλ‘, CCE(D)κ° κ΅¬κ°κ·Έλνκ° λκΈ° μν μ±λΆμ κ°μμ λν μΆ©λΆμ‘°κ±΄μ ꡬνμλ€.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) 170λ°
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