Intersection Representation Research Articles

Local and union boxicity

The boxicity box(H) of a graph H is the smallest integer d such that H is the intersection of d interval graphs, or equivalently, that H is the intersection graph of axis-aligned boxes in Rd. These intersection representations can be interpreted as covering representations of the complement Hc of H with co-interval graphs, that is, complements of interval graphs. We follow the recent framework of global, local and folded covering numbers (Knauer and Ueckerdt, 2016) to define two new parameters: the local boxicity boxℓ(H) and the union boxicity box¯(H) of H. The union boxicity of H is the smallest d such that Hc can be covered with d vertex–disjoint unions of co-interval graphs, while the local boxicity of H is the smallest d such that Hc can be covered with co-interval graphs, at most d at every vertex.We show that for every graph H we have boxℓ(H)≤box¯(H)≤box(H) and that each of these inequalities can be arbitrarily far apart. Moreover, we show that local and union boxicity are also characterized by intersection representations of appropriate axis-aligned boxes in Rd. We demonstrate with a few striking examples, that in a sense, the local boxicity is a better indication for the complexity of a graph, than the classical boxicity.

Sublinear approximation algorithms for boxicity and related problems

The boxicity of a graph G(V,E) is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes in Rk. Cubicity is a variant of boxicity, where the axis parallel boxes in the intersection representation are restricted to be of unit length sides. Deciding whether the boxicity (resp. cubicity) of a graph is at most k is NP-hard, even for k=2 or 3. Computing these parameters is inapproximable within O(n1−ϵ)-factor, for any ϵ>0 in polynomial time unless NP=ZPP, even for many simple graph classes.In this paper, we give a polynomial time κ(n) factor approximation algorithm for computing boxicity and a κ(n)⌈loglogn⌉ factor approximation algorithm for computing the cubicity, where κ(n)=2nloglogn∕logn. These o(n) factor approximation algorithms also produce the corresponding box (resp. cube) representations. As a special case, this resolves the question posed by Spinrad (2003) about polynomial time construction of o(n) dimensional box representations for boxicity 2 graphs. Other consequences of our approximation algorithm include O(κ(n)) factor approximation algorithms for computing the following parameters: the partial order dimension (poset dimension) of finite posets, the interval dimension of finite posets, minimum chain cover of bipartite graphs, Ferrers dimension of digraphs and threshold dimension of split graphs and co-bipartite graphs. Each of these parameters is inapproximable within an O(n1−ϵ)-factor, for any ϵ>0 in polynomial time unless NP=ZPP and the algorithms we derive seem to be the first o(n) factor approximation algorithms known for all these problems. We note that obtaining a o(n) factor approximation for poset dimension was also mentioned as an open problem by Felsner et al. (2017).In the second part of this paper, parameterized approximation algorithms for boxicity using various edit distance parameters are derived. We also present a parameterized approximation scheme for cubicity, using minimum vertex cover number as the parameter.

Latent Network Features and Overlapping Community Discovery via Boolean Intersection Representations.

We propose a new latent Boolean feature model for complex networks that captures different types of node interactions and network communities. The model is based on a new concept in graph theory, termed the Boolean intersection representation of a graph, which generalizes the notion of an intersection representation. We mostly focus on one form of Boolean intersection, termed cointersection, and describe how to use this representation to deduce node feature sets and their communities. We derive several general bounds on the minimum number of features used in cointersection representations and discuss graph families for which exact cointersection characterizations are possible. Our results also include algorithms for finding optimal and approximate cointersection representations of a graph.

“Older-wiser-lesbians” and “baby-dykes”: mediating age and generation in New Queer Cinema

Representations of intersections of gender, age, and sexuality can reveal deep-rooted cultural anxieties about older women and sexuality. Images of lesbian ageing are of particular interest in terms of alterity, as the old/er queer woman can combine layers of otherness—not only is she the cultural “other” within heteronormativity, but she can also appear as the opposite of popular culture’s lesbian chic. In this article, a cultural analysis of a range of films—If These Walls Could Talk 2 (dir. Anderson, Coolidge, and Heche 2000), Itty Bitty Titty Committee (dir. Babbit 2007), The Owls (dir. Dunye 2010), Hannah Free (dir. Carlton 2009), and Cloudburst (dir. Fitzgerald 2011)—considers diverse dramatisations of lesbian generations. This article interrogates to what extent alternative cinemas deconstruct normative conceptualisations of ageing. Drawing on recent critiques of post-feminist culture, and a range of feminist and age/ing studies scholarship, it suggests that a linear understanding of ageing and the generational underlies dominant depictions of oppositional binaries of young versus old, of generational segregation or rivalry, and the othering of age. It concludes that non-linear understandings of temporality and ageing contain the potential for New Queer Cinema to counteract such idealisations of youthfulness, which, it argues, is one of the most deep-rooted manifestations of (hetero)normativity.

On String Graph Limits and the Structure of a Typical String Graph

AbstractWe study limits of convergent sequences of string graphs, that is graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We also prove similar results for several related graph classes.

Intersection graphs of L-shapes and segments in the plane

An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: ▪, ▪,▪ and ▪. A k-bend path is a simple path in the plane, whose direction changes k times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an ▪, an ▪ or ▪, a k-bend path, or a segment, then this graph is called an {▪}-graph, {▪,▪}-graph, Bk-VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer (1992), stating that every {▪,▪}-graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are {▪}-graphs, or Bk-VPG-graphs for some small constant k. We show that all planar 3-trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are {▪}-graphs. Furthermore we show that complements of planar graphs are B17-VPG-graphs and complements of full subdivisions are B2-VPG-graphs. Here a full subdivision is a graph in which each edge is subdivided at least once.

De-constructing monoculturalism on the German screen: A critical cultural reading of On the Other Side

Abstract This article provides a critical cultural analysis of Fatih Akin’s On the Other Side; a celebrated European drama and a multiple award winner, as a case study depicting contemporary German Turkish intercultural dynamics. It focuses on the politics of intersectional representation of Turks and examines implications of strategic cinematographic othering in today’s Germany, marked by growing Islamophobia. Building upon Chancellor Merkel’s 2010 statement about the death of German multiculturalism, as well as her 2015 statement about multiculturalism being a ‘White Lie’, this article introduces a concept of Monoculturalism and explores its on-screen manifestations and cultural implications for the contemporary Europe.

The “Other” Woman in Contemporary Television Drama: Analyzing Intersectional Representation on Bones

This essay theorizes intersectional representation as a strategic device used in twenty-first century television narrative by examining the generically “Other” woman in television drama. This character is often racially, ethnically, economically and sexually “othered”. She appears on numerous broadcast television programs, and yet remains relatively invisible in cultural discourses of television. I offer a critical reading of the intersectional representation used on FOX’s series Bones as a means of exposing how intersectional imagery functions to re-center identity politics on White, heteronormative, middle-class values. That these images remain largely invisible in academic discussions of race, ethnicity, sexuality and class on television while appearing frequently in the most economically successful television narratives speaks to the complexity of intersectional identity politics in contemporary media studies.

Gender Quotas and Ethnic Minority Representation: Swedish Evidence from a Longitudinal Mixed Methods Study

In this paper, we study the ways in which affirmative action for one political minority, gender quotas, impact on intersectional representation. In a quantitative analysis of detailed panel data from 285 Swedish municipal assemblies, the numerical impact of a zipper placement mandate in Sweden's largest political party, the Social Democratic Party, is analyzed. No evidence that this quota helped, or hindered, the intersectional representation of men or women is found in the short run, but it is found that a weak numerical impact may exist in the long run. A qualitative analysis of party records and interviews with key actors sheds further light on these results. Differences in the norms of representation for women and polyethnic minorities, coupled with weak organizational and practical constraints for formulating policies for the latter, appear to be likely explanations.

Mathematical algorithm development and parametric studies with the GEOFRAC three-dimensional stochastic model of natural rock fracture systems

This paper presents results from research conducted at MIT during 2010–2012 on modeling of natural rock fracture systems with the GEOFRAC three-dimensional stochastic model. Following a background summary of discrete fracture network models and a brief introduction of GEOFRAC, the paper provides a thorough description of the newly developed mathematical and computer algorithms for fracture intensity, aperture, and intersection representation, which have been implemented in MATLAB. The new methods optimize, in particular, the representation of fracture intensity in terms of cumulative fracture area per unit volume, P32, via the Poisson-Voronoi Tessellation of planes into polygonal fracture shapes. In addition, fracture apertures now can be represented probabilistically or deterministically whereas the newly implemented intersection algorithms allow for computing discrete pathways of interconnected fractures. In conclusion, results from a statistical parametric study, which was conducted with the enhanced GEOFRAC model and the new MATLAB-based Monte Carlo simulation program FRACSIM, demonstrate how fracture intensity, size, and orientations influence fracture connectivity.

Gendering Creolisation: Creolising Affect

Going beyond the creolisation theories of Brathwaite and Glissant, I attempt to develop ideas concerning the gendering of creolisation, and a historicising of affects within it. Addressing affects as ‘physiological things’ contextualised in the history of the Caribbean slave plantation, I seek, importantly, to delineate a trajectory and development of a specific Creole history in relation to affects. Brathwaite's proposition that ‘the most significant (and lasting) inter-cultural creolisation took place’ within the ‘intimate’ space of ‘sexual relations’ is key to my argument. In the light of this, I consider how Creole cultural knowledge about affect—as the primary motivational system inclusive of fear, anger, outrage and so on — might be identified, and what constitutes such Creole knowledge within which affect might be embedded. How might Glissant's relationality and Creole literary texts add to this knowledge? I focus primarily on three texts: Clarke's The Polished Hoe, Collins’ The Colour of Forgetting and Morejón's ‘Amo a mi amo’/‘l Love My Master’. Each text on which I draw is selected for its intersectional representation of gender relations, ‘knowledge about sexual difference’ and its representation of Creole subjectivities within the context problematised here as the ‘demonic ground’. Moreover, as auto-theorising texts, they represent both narrative and meta-narrative of a creolising of affects in and against the economy of the slave plantation. Each represents also a stage or aspect of the development of subjectivities and an affective community that inform this intervention concerned with theorising against consolidated, universalising and Eurocentric conceptualisations of affect. In the process, I attempt to offer a differentiated cartography and literary archaeology of affect.

Tractabilities and Intractabilities on Geometric Intersection Graphs

A graph is said to be an intersection graph if there is a set of objects such that each vertex corresponds to an object and two vertices are adjacent if and only if the corresponding objects have a nonempty intersection. There are several natural graph classes that have geometric intersection representations. The geometric representations sometimes help to prove tractability/intractability of problems on graph classes. In this paper, we show some results proved by using geometric representations.

Planar Graphs as VPG-Graphs

A graph is Bk-VPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2-VPG. We also show that the 4-connected planar graphs are a subclass of the intersection graphs of Z-shapes (i.e., a special case of B2-VPG). Additionally, we demonstrate that a B2-VPG representation of a planar graph can be constructed in O(n3/2) time. We further show that the triangle-free planar graphs are contact graphs of: L-shapes, Γ-shapes, vertical segments, and horizontal segments (i.e., a special case of contact B1-VPG). From this proof we gain a new proof that bipartite planar graphs are a subclass of 2-DIR.

On Minimal Fuzzy Ideals of Semigroups

The present paper contains the sufficient condition of a fuzzy semigroup to be a fuzzy group using fuzzy points. The existence of a fuzzy kernel in semigroup is explored. It has been shown that every fuzzy ideal of a semigroup contains every minimal fuzzy left and every minimal fuzzy right ideal of semigroup. The fuzzy kernel is the class sum of minimal fuzzy left (right) ideals of a semigroup. Every fuzzy left ideal of a fuzzy kernel is also a fuzzy left ideal of a semigroup. It has been shown that the product of minimal fuzzy left ideal and minimal fuzzy right ideal of a semigroup forms a group. The representation of minimal fuzzy left (right) ideals and also the representation of intersection of minimal fuzzy left ideal and minimal fuzzy right ideal are shown. The fuzzy kernel of a semigroup is basically the class sum of all the minimal fuzzy left (right) ideals of a semigroup. Finally the sufficient condition of fuzzy kernel to be completely simple semigroup has been proved.

Integral mixed unit interval graphs

We characterize graphs that have intersection representations using unit intervals with open or closed ends such that all ends of the intervals are integral in terms of infinitely many minimal forbidden induced subgraphs. Furthermore, we provide a linear-time algorithm that decides if a given interval graph admits such an intersection representation.

Mixed unit interval graphs

The class of intersection graphs of unit intervals of the real line whose ends may be open or closed is a strict superclass of the well-known class of unit interval graphs. We pose a conjecture concerning characterizations of such mixed unit interval graphs, verify parts of it in general, and prove it completely for diamond-free graphs. In particular, we characterize diamond-free mixed unit interval graphs by means of an infinite family of forbidden induced subgraphs, and we show that a diamond-free graph is mixed unit interval if and only if it has intersection representations using unit intervals such that all ends of the intervals are integral.

Unit and single point interval graphs

We describe a linear time algorithm for the recognition of graphs that have an intersection representation using unit length intervals and single point intervals. Furthermore, we characterize these graphs using forbidden induced subgraphs.

MINIMUM WEIGHT FEEDBACK VERTEX SETS IN CIRCLE n-GON GRAPHS AND CIRCLE TRAPEZOID GRAPHS

A circle n-gon is the region between n or fewer non-crossing chords of a circle, no chord connecting the arcs between two other chords; the sides of a circle n-gon are either chords or arcs of the circle. A circle n-gon graph is the intersection graph of a family of circle n-gons in a circle. The family of circle trapezoid graphs is exactly the family of circle 2-gon graphs and the family of circle graphs is exactly the family of circle 1-gon graphs. The family of circle n-gon graphs contains the polygon-circle graphs which have an intersection representation by circle polygons, each polygon with at most n chords. We describe a polynomial time algorithm to find a minimum weight feedback vertex set, or equivalently, a maximum weight induced forest, in a circle n-gon graph with positive weights, when its intersection model by n-gon-interval-filaments is given.

Dimension-2 poset competition numbers and dimension-2 poset double competition numbers

Let D = ( V ( D ) , A ( D ) ) be a digraph. The competition graph of D , is the graph with vertex set V ( D ) and edge set { u v ∈ V ( D ) 2 : ∃ w ∈ V ( D ) , u w ⃗ , v w ⃗ ∈ A ( D ) } . The double competition graph of D , is the graph with vertex set V ( D ) and edge set { u v ∈ V ( D ) 2 : ∃ w 1 , w 2 ∈ V ( D ) , u w 1 ⃗ , v w 1 ⃗ , w 2 u ⃗ , w 2 v ⃗ ∈ A ( D ) } . A poset of dimension at most two is a digraph whose vertices are some points in the Euclidean plane R 2 and there is an arc going from a vertex ( x 1 , y 1 ) to a vertex ( x 2 , y 2 ) if and only if x 1 > x 2 and y 1 > y 2 . We show that a graph is the competition graph of a poset of dimension at most two if and only if it is an interval graph, at least half of whose maximal cliques are isolated vertices. This answers an open question on the doubly partial order competition number posed by Cho and Kim. We prove that the double competition graph of a poset of dimension at most two must be a trapezoid graph, generalizing a result of Kim, Kim, and Rho. Some connections are also established between the minimum numbers of isolated vertices required to be added to change a given graph into the competition graph, the double competition graph, of a poset and the minimum sizes of certain intersection representations of that graph.

“I’m Just Trying to Find my Way Like Most Kids”: Bisexuality, Adolescence and the Drama of One Tree Hill

This essay offers a narrative reading of the representation of bisexuality on One Tree Hill by examining the character Anna Tagaro. Grounding this reading in observations about bisexuality, media representation and adolescent identity formation processes, the essay exposes Anna’s representation as both a viable coming out story for an adolescent audience and a systematic erasure of bisexuality as a valid social identity. The displacement of political activism with friend and ally Peyton creates a representation that functions both as liberating and constraining simultaneously. Moreover, Anna’s inclusion as the only Latina character in an all white, all heterosexual cast offers an intersectional representation of race and sexual identity. This conflict between progress and constraint in the representation of youth identity choices offers scholars ample data for future studies in teen television and sexuality.