Forbidden Terms Research Articles

FORBIDDEN INDUCED SUBGRAPHS AND THE ŁOŚ–TARSKI THEOREM

AbstractLet $\mathscr {C}$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Łoś–Tarski Theorem from classical model theory implies that $\mathscr {C}$ is definable in first-order logic by a sentence $\varphi $ if and only if $\mathscr {C}$ has a finite set of forbidden induced finite subgraphs. This result provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms of forbidden induced finite subgraphs. Furthermore, by the Completeness Theorem, we can compute from $\varphi $ the corresponding forbidden induced subgraphs. This machinery fails on finite graphs as shown by our results: –There is a class $\mathscr {C}$ of finite graphs that is definable in first-order logic and closed under induced subgraphs but has no finite set of forbidden induced subgraphs.–Even if we only consider classes $\mathscr {C}$ of finite graphs that can be characterized by a finite set of forbidden induced subgraphs, such a characterization cannot be computed from a first-order sentence $\varphi $ that defines $\mathscr {C}$ and the size of the characterization cannot be bounded by $f(|\varphi |)$ for any computable function f.Besides their importance in graph theory, the above results also significantly strengthen similar known theorems for arbitrary structures.

A characterization of unit interval bigraphs of open and closed intervals

Interval graphs are the intersection graphs of intervals on the real line. Unit interval graphs are interval graphs representable by intervals of unit length. Rautenbach and Szwarcfiter showed that the class of intersection graphs of the open and closed real intervals of unit length is a strict superclass of the class of unit interval graphs. They also characterized this class of graphs.An interval bigraph is a bipartite graph where to each vertex we can assign an interval so that vertices in the opposite partite sets are adjacent if and only if their intervals intersect. Analogously we show that the class of finite intersection bigraphs of open and closed intervals of unit length is a strict superclass of the class of unit interval bigraphs. We also characterize this class of bigraphs in terms of forbidden induced subgraphs and a suitable extension of proper interval bigraphs.

프랑스 선교사들의 식민지 사제 양성

During the Japanese colonial period, the training of priests in the Korean Catholic Church was a long-term program that spanned over 12 years of education, giving rise to a distinctive group of intellectuals during the colonial era. This distinctiveness was further compounded by the conflict and confusion between modernity and anti-modernity, with the defense mechanisms of French missionaries' anti-modernism and anti-republican ideology being integrated into seminary education.
 The Japanese colonial education policy, which undermined advanced education by emphasizing the so-called 'harm of reading', drew exquisite agreement from missionaries who were proud of the revolution and republic but internalized anger against secular and anti-church values. The missionaries accepted the 'accreditation' system while tactfully retaining the church's values in actual education, and dealt with it with a backside disagreement.
 The so-called fervor for modern education and the legacy of liberalism stirred unrest among the youth. The notion that it eventually fostered socialism posed an equal risk perceived by both the missionary groups and colonial authorities, leading to a reluctance towards higher education. Requests for the establishment of a united theological seminary and the reform of existing seminaries, even those demanded at the Vatican level,were stymied as they were relegated to minor seminary-level secondary education accreditation. The accreditation for the theological college, which boasted the finest faculty for priestly training, was ignored.
 The bishops swiftly countered the burgeoning liberalism as a 'negative influence' and promptly removed it from the seminaries. However, even amidst the tumultuous clash of modernity and anti- modernity, the awakened consciousness itself couldn't be extinguished. The exiled republican government in Shanghai, formed by the March 1st Movement, finally unveiled the forbidden terms 'republic' and 'civilization' to Korean priests.

Obstructions for χ-diperfectness

In 1982, Berge defined the class of χ-diperfect digraphs. A digraph D is χ-diperfect if for every minimum coloring S of D there is a path P containing exactly one vertex of each color class of S and this property holds for every induced subdigraph of D. The ultimate goal in this research area is to obtain a characterization of χ-diperfect digraphs in terms of forbidden induced subdigraphs, but this may be a very difficult problem and not likely to be solved in a near future. Berge showed the first examples of obstructions for χ-diperfect digraphs (i.e. minimal non-χ-diperfect digraphs) by presenting orientations of odd cycles and complements of odd cycles that are not χ-diperfect. In 2022, de Paula Silva, Nunes da Silva and Lee showed characterizations of non-χ-diperfect super-orientations of odd cycles and their complements. Moreover, they showed that these structures are not the only obstructions for χ-diperfect digraphs, by presenting new obstructions with stability number two and three. In this paper, we present new obstructions for χ-diperfect digraphs with arbitrary stability number and arbitrary chromatic number.

Forbidden subgraphs in generating graphs of finite groups

Let $G$ be a $2$-generated group. The generating graph $\Gamma(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g_1$ and $g_2$ are adjacent if $G = \langle g_1, g_2 \rangle.$ This graph encodes the combinatorial structure of the distribution of generating pairs across $G.$ In this paper we study some graph theoretic properties of $\Gamma(G)$, with particular emphasis on those properties that can be formulated in terms of forbidden induced subgraphs. In particular we investigate when the generating graph $\Gamma(G)$ is a cograph (giving a complete description when $G$ is soluble) and when it is perfect (giving a complete description when $G$ is nilpotent and proving, among the others, that $\Gamma(S_n)$ and $\Gamma(A_n)$ are perfect if and only if $n\leq 4$). Finally we prove that for a finite group $G$, the properties that $\Gamma(G)$ is split, chordal or $C_4$-free are equivalent.

Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs I: Nets and bulls

AbstractIn a series of papers, of which this is the first, we study sufficient conditions for Hamiltonicity in terms of forbidden induced subgraphs and extend such results to locally finite infinite graphs. For this we use topological circles within the Freudenthal compactification of a locally finite graph as infinite cycles. In this paper we focus on conditions involving claws, nets and bulls as induced subgraphs. We extend Hamiltonicity results for finite claw‐free and net‐free graphs by Shepherd to locally finite graphs. Moreover, we generalise a classification of finite claw‐free and net‐free graphs by Shepherd to locally finite ones. Finally, we extend to locally finite graphs a Hamiltonicity result by Ryjáček involving a relaxed condition of being bull‐free.

Upward-closed hereditary families in the dominance order

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Ruch and Gutman (1979) and Merris (2002), the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class $\mathcal{F}$ of graphs to be dominance monotone if whenever no realization of $e$ contains an element $\mathcal{F}$ as an induced subgraph, and $d$ majorizes $e$, then no realization of $d$ induces an element of $\mathcal{F}$. We present conditions necessary for a set of graphs to be dominance monotone, and we identify the dominance monotone sets of order at most 3.

Forbidden Subgraphs of Power Graphs

The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$.
 A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.

Two classes of [formula omitted]-perfect graphs that do not necessarily have simplicial extremes

For a graph G, let β(G)=max{δ(G′)+1∣G′ is an induced subgraph of G}. This parameter is an upper bound on the chromatic number of a graph. A graph G is β-perfect if χ(G′)=β(G′) for all induced subgraphs G′ of G. A number of classes have been shown in literature to be β-perfect, but for the class of all β-perfect graphs, the complexity of their recognition and characterization in terms of forbidden induced subgraphs remain open.It is known that minimally β-imperfect graphs cannot have a simplicial extreme, i.e. a vertex whose neighborhood is a clique or of size 2. β-perfection of all the known β-perfect classes of graphs was shown through the existence of simplicial extremes. In this paper we study two classes of β-perfect graphs that do not necessarily have this property.Firstly, we prove that (even hole, twin wheel, cap)-free graphs are β-perfect. This class properly contains chordal graphs, and is the only known generalization of chordal graphs that is shown to be β-perfect.Secondly, we give a complete structural characterization of β-perfect hyperholes (graphs obtained from a hole by a sequence of clique substitutions), which we then use to give a linear-time algorithm to recognize whether a hyperhole is β-perfect. We also obtain a complete list of minimally β-imperfect hyperholes.

Polynomial $$\chi $$ χ -Binding Functions and Forbidden Induced Subgraphs: A Survey

A graph G with clique number $$\omega (G)$$ and chromatic number $$\chi (G)$$ is perfect if $$\chi (H)=\omega (H)$$ for every induced subgraph H of G. A family $${\mathcal {G}}$$ of graphs is called $$\chi $$ -bounded with binding function f if $$\chi (G') \le f(\omega (G'))$$ holds whenever $$G \in {\mathcal {G}}$$ and $$G'$$ is an induced subgraph of G. In this paper we will present a survey on polynomial $$\chi $$ -binding functions. Especially we will address perfect graphs, hereditary graphs satisfying the Vizing bound ( $$\chi \le \omega +1$$ ), graphs having linear $$\chi $$ -binding functions and graphs having non-linear polynomial $$\chi $$ -binding functions. Thereby we also survey polynomial $$\chi $$ -binding functions for several graph classes defined in terms of forbidden induced subgraphs, among them $$2K_2$$ -free graphs, $$P_k$$ -free graphs, claw-free graphs, and $${ diamond}$$ -free graphs. ( [])

On the 12-representability of induced subgraphs of a grid graph

The notion of a 12-representable graph was introduced by Jones et al.. This notion generalizes the notions of the much studied permutation graphs and co-interval graphs. It is known that any 12-representable graph is a comparability graph, and also that a tree is 12-representable if and only if it is a double caterpillar. Moreover, Jones et al. initiated the study of 12-representability of induced subgraphs of a grid graph, and asked whether it is possible to characterize such graphs. This question in is meant to be about induced subgraphs of a grid graph that consist of squares, which we call square grid graphs. However, an induced subgraph in a grid graph does not have to contain entire squares, and we call such graphs line grid graphs. In this paper we answer the question of Jones et al. by providing a complete characterization of $12$-representable square grid graphs in terms of forbidden induced subgraphs. Moreover, we conjecture such a characterization for the line grid graphs and give a number of results towards solving this challenging conjecture. Our results are a major step in the direction of characterization of all 12-representable graphs since beyond our characterization, we also discuss relations between graph labelings and 12-representability, one of the key open questions in the area.

Formula omitted]-perfectly orientable [formula omitted]-minor-free and outerplanar graphs

A graph G is said to be 1-perfectly orientable if it has an orientation D such that for every vertex v∈V(G), the out-neighborhood of v in D is a clique in G. In 1982, Skrien posed the problem of characterizing the class of 1-perfectly orientable graphs. This graph class forms a common generalization of the classes of chordal and circular arc graphs; however, while polynomially recognizable via a reduction to 2-SAT, no structural characterization of this intriguing class of graphs is known. Based on a reduction of the study of 1-perfectly orientable graphs to the biconnected case, we characterize, both in terms of forbidden induced minors and in terms of composition theorems, the classes of 1-perfectly orientable K4-minor-free graphs and of 1-perfectly orientable outerplanar graphs. As part of our approach, we introduce a class of graphs defined similarly as the class of 2-trees and relate the classes of graphs under consideration to two other graph classes closed under induced minors studied in the literature: cyclically orientable graphs and graphs of separability at most 2.

Global cycle properties in graphs with large minimum clustering coefficient

Let 𝒫 be a graph property. A graph G is said to be locally 𝒫 (closed locally 𝒫) if the subgraph induced by the open neighbourhood (closed neighbourhood, respectively) of every vertex in G has property 𝒫. The clustering coefficient of a vertex is the proportion of pairs of its neighbours that are themselves neighbours. The minimum clustering coefficient of G is the smallest clustering coefficient among all vertices of G. Let H be a subgraph of a graph G and let S ⊆ V (H). We say that H is a strongly induced subgraph of G with attachment set S, if H is an induced subgraph of G and the vertices of V (H) − S are not incident with edges that are not in H. A graph G is fully cycle extendable if every vertex of G lies in a triangle and for every nonhamiltonian cycle C of G, there is a cycle of length |V (C)| + 1 that contains the vertices of C. A complete characterization, of those locally connected graphs with minimum clustering coefficient 1/2 and maximum degree at most 6 that are fully cycle extendable, is given in terms of forbidden strongly induced subgraphs (with specified attachment sets). Moreover, it is shown that all locally connected graphs with Δ ≤ 6 and sufficiently large minimum clustering coefficient are weakly pancylic, thereby proving Ryj´ǎcek’s conjecture for this class of graphs.

A note on an induced subgraph characterization of domination perfect graphs

Let γ(G) and ι(G) be the domination and independent domination numbers of a graph G, respectively. Introduced by Sumner and Moore (1979), a graph G is domination perfect if γ(H)=ι(H) for every induced subgraph H⊆G. In 1991, Zverovich and Zverovich proposed a characterization of domination perfect graphs in terms of forbidden induced subgraphs. Fulman (1993) noticed that this characterization is not correct. Later, Zverovich and Zverovich (1995) offered such a second characterization with 17 forbidden induced subgraphs. However, the latter still needs to be adjusted.In this paper, we point out a counterexample. We then give a new characterization of domination perfect graphs in terms of only 8 forbidden induced subgraphs and a short proof thereof. Moreover, in the class of domination perfect graphs, we propose a polynomial-time algorithm computing, given a dominating set D, an independent dominating set Y such that |Y|≤∣D∣.

Graphs with maximal induced matchings of the same size

A graph is well-indumatched if all its maximal induced matchings are of the same size. We first prove that recognizing whether a graph is well-indumatched is a co-NP-complete problem even for (2P5,K1,5)-free graphs. We then show that decision problems Independent Dominating Set, Independent Set, and Dominating Set are NP-complete for the class of well-indumatched graphs. We also show that this class is a co-indumatching hereditary class, i.e., it is closed under deleting the end-vertices of an induced matching along with their neighborhoods, and we characterize well-indumatched graphs in terms of forbidden co-indumatching subgraphs. We prove that recognizing a co-indumatching subgraph is an NP-complete problem. We introduce a perfectly well-indumatched graph, in which every induced subgraph is well-indumatched, and characterize the class of these graphs in terms of forbidden induced subgraphs. Finally, we show that the weighted versions of problems Independent Dominating Set and Independent Set can be solved in polynomial time for perfectly well-indumatched graphs, but problem Dominating Set is NP-complete.

Ti esti state-terrorism?

Este trabajo aborda la cuestión filosófica y urgente del “ti esti state-terrorism”. El ensayo analiza críticamente y filosóficamente el concepto del terrorismo del estado, basado en una discusión epistemológica, que intenta re-formular la cuestión del terrorismo, donde se argumenta que el terrorismo de estado es el fenómeno inicial y la raíz del terrorismo en todas sus formas. Mi aspiración es contribuir a una comprensión del terrorismo que trate de revelar cómo en griego –el idioma de los conceptos y la filosofía– el término “terrorismo” es “terror-cracia” y significa literalmente el terrorismo del estado; también es parte de la misma familia de la democracia, al tener el sufijo “–cracia.” Este reconocimiento puede revelar nuevas maneras de ver el concepto y fenómeno del terrorismo, o incluso puede ayudar a redefinir el concepto por completo al ponerlo en sintonía con su sentido original. Según este razonamiento, el terrorismo se referiría a una forma del gobierno y a un sistema político. En mi ensayo introduzco esta hipótesis al hilo de diálogos filosóficos sobre el terrorismo, ya que el asunto no se había discutido nunca antes en estos términos. “El terrorismo del estado” es un término prohibido según los expertos jurídicos, pero los pensadores críticos no deben tratarlo como término estéril.

Clique cycle-transversals in distance-hereditary graphs

A cycle transversal or feedback vertex set of a graph G is a subset T⊆V(G) such that T∩V(C)≠0̸ for every cycle C of G. A clique cycle transversal, or cct for short, is a cycle transversal which is a clique. Recognizing graphs which admit a cct can be done in polynomial time; however, no structural characterization of such graphs is known. We characterize distance-hereditary graphs admitting a cct in terms of forbidden induced subgraphs. This extends similar results for chordal graphs and cographs.

On a class of graphs between threshold and total domishold graphs

A total dominating set in a graph is a subset of vertices such that every vertex in the graph has a neighbor in it. A graph is said to be total domishold if it admits a total domishold structure, that is, a hyperplane with non-negative coefficients that separates the characteristic vectors of its total dominating sets from the characteristic vectors of other vertex subsets. Hereditary total domishold graphs, that is, graphs every induced subgraph of which is total domishold, were recently characterized in terms of forbidden induced subgraphs; this characterization leads to an O(|V(G)|6) recognition algorithm of hereditary total domishold graphs.In this paper, we study a subclass G of the hereditary total domishold graphs obtained by replacing two of the forbidden induced subgraphs of order 6 with two of their proper induced subgraphs of order 5. We show that every connected graph in G is a leaf extension of a threshold graph, that is, it can be obtained from some threshold graph G by attaching some pendant vertices to each vertex of G. Using this property, we develop a structural characterization of graphs in G, which implies a linear time recognition algorithm for graphs in this class. We also give a linear time algorithm for computing a total domishold structure of a graph in G. In contrast to the algorithm for the general case of (hereditary) total domishold graphs, which relies on solving a linear program, our algorithm is purely combinatorial and works directly with the graph.

Clique cycle transversals in distance-hereditary graphs

A cycle transversal of a graph G is a subset T⊆V(G) such that T∩V(C)≠∅ for every cycle C of G. A clique cycle transversal, or cct for short, is a cycle transversal which is a clique. Recognizing graphs which admit a cct can be done in polynomial time; however, no structural characterization of such graphs is known. We characterize distance-hereditary graphs admitting a cct in terms of forbidden induced subgraphs. This extends similar results for chordal graphs and cographs.

A forbidden subgraph characterization of some graph classes using betweenness axioms

Let IG(x,y) and JG(x,y) be the geodesic and induced path intervals between x and y in a connected graph G, respectively. The following three betweenness axioms are considered for a set V and R:V×V→2V: (i)x∈R(u,y),y∈R(x,v),x≠y,|R(u,v)|>2⇒x∈R(u,v);(ii)x∈R(u,v)⇒R(u,x)∩R(x,v)={x};(iii)x∈R(u,y),y∈R(x,v),x≠y⇒x∈R(u,v). We characterize the class of graphs for which IG satisfies (i), the class for which JG satisfies (ii) and the class for which IG or JG satisfies (iii). The characterization is given in terms of forbidden induced subgraphs. It turns out that the class of graphs for which IG satisfies (i) is a proper subclass of distance hereditary graphs and the class for which JG satisfies (ii) is a proper superclass of distance hereditary graphs. We also give an axiomatic characterization of chordal and Ptolemaic graphs.