# non-word representable graphs

## Top PDF non-word representable graphs: ### Word-representability of triangulations of grid-covered cylinder graphs

Our proof is organized as follows. In Subsection 4.1 we will provide all six minimum non-word-representable graphs that can appear in triangulations of GCCGs with three sectors (see Figure 4.11) and give an explicit proof that one of these graphs is non-word-representable. Then, in Subsection 4.2, we will give an inductive argument showing that avoidance of the six graphs in Figure 4.11 is a sufficient condition for a GCCG with three sectors to be word-representable. Note that the graphs in Figure 4.11 were obtained by an exhaustive computer search on graphs on up to eight vertices. However, our argument in Subsection 4.2 will show that no other non-word-representable induced subgraphs can be found among all triangulations of GCCGs with three sectors. ### On k-11-representable graphs

The paper is organized as follows. In the rest of the section, we give more details about word-representable graphs. In Section 2, we introduce rigorously the notion of a k-11- representable graph and provide a number of general results on these graphs. In particular, we show that a (k − 1)-11-representable graph is necessarily k-11-representable (see Theo- rem 2.2). In Section 3, we study the class of 1-11-representable graphs. These studies are extended in Section 4, where we 1-11-represent all non-word-representable graphs on at most 7 vertices. In Section 5 we prove that any graph is 2-11-representable. Finally, in Section 6, we state a number of open problems on k-11-representable graphs. ### Solving computational problems in the theory of word-representable graphs

It is also interesting to identify minimal non-word-representable graphs of each size, i.e. graphs containing no non-word-representable strict induced subgraphs. To do this, we stored all non-word-representable graphs of each size. After computing with geng all possible graphs with one more vertex, we eliminate graphs containing one of the stored graphs as an induced subgraph. We did this with a simple constraint model which tries to find a mapping from the vertices of the induced subgraph to the vertices of the larger graph, and if successful discards the larger graph from consideration. This enabled us to count all minimal non- word-representable graphs of each size up to 9, which is shown in Table 2. The filtering process we used was too inefficient to complete the cases n ≥ 10. ### Solving computational problems in the theory of word representable graphs

It is also interesting to identify minimal non-word-representable graphs of each size, i.e. graphs containing no non-word-representable strict induced subgraphs. To do this, we stored all non-word-representable graphs of each size. After computing with geng all possible graphs with one more vertex, we eliminate graphs containing one of the stored graphs as an induced subgraph. We did this with a simple constraint model which tries to find a mapping from the vertices of the induced subgraph to the vertices of the larger graph, and if successful discards the larger graph from consideration. This enabled us to count all minimal non- word-representable graphs of each size up to 9, which is shown in Table 2. The filtering process we used was too inefficient to complete the cases n ≥ 10. ### Semi-transitive orientations and word-representable graphs

Related work. The notion of directed word-representable graphs was in- troduced in  to obtain asymptotic bounds on the free spectrum of the widely-studied Perkins semigroup, which has played central role in semi- group theory since 1960, particularly as a source of examples and coun- terexamples. In , numerous properties of word-representable graphs were derived and several types of word-representable and non-word-representable graphs pinpointed. Some open questions from  were resolved recently in , including the representability of the Petersen graph. ### Existence of μ-representation of graphs

The notion of a word-representable graph first appears in . A simple graph G = (V, E) is word-representable if there exists a word w over the alphabet V such that letters x and y, x ̸ = y, alternate in w if and only if xy is an edge in E. By definition, each letter in V must occurs at least once in w. For example, the cycle graph on four vertices labeled by 1, 2, 3 and 4 in clockwise direction can be represented by the word 14213243. Some graphs are word-representable, the others are not; the minimum non-word-representable graph is the wheel W 5 ### Word Representability of Line Graphs

A graph G =  V E ,  is representable if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if   x y , is in E for each x not equal to . The motivation to study representable graphs came from algebra, but this subject is interesting from graph theoretical, com- puter science, and combinatorics on words points of view. In this paper, we prove that for greater than 3, the line graph of an -wheel is non-representable. This not only provides a new construction of non-repre- sentable graphs, but also answers an open question on representability of the line graph of the 5-wheel, the minimal non-representable graph. Moreover, we show that for greater than 4, the line graph of the com- plete graph is also non-representable. We then use these facts to prove that given a graph which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of -times is guaranteed to be non-representable for greater than 3. ### Representing graphs via pattern avoiding words

7. Other notions of word-representable graphs As it is mentioned in Section 2, apart from our main generalization, given in Definition 1, of the notion of a word-representable graph, we have another generalization given in Definition 5 below. In this section, we also state some other ways to define the notion of a (directed) graph representable by words. Our definitions can be generalized to the case of hypergraphs by simply allowing words defining edges/non-edges be over alphabets containing more than two letters. However, the focus of this paper was studying 12-representable graphs, so we leave all the notions introduced below for a later day to study. ### New results on word-representable graphs

Distinct letters x and y alternate in a word w if the deletion of all other letters from the word results in either xyxy · · · or yxyx · · · . A graph G = (V, E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if (x, y) ∈ E for each x 6= y . For example, the graph M in Figure 1 is word-representable, because the word w = 1213423 has the right alternating properties, i.e. the only non-alternating pairs in this word are 1,3 and 1,4 that correspond to the only non-adjacent pairs of vertices in the graph. ### Word-representability of face subdivisions of triangular grid graphs

The triangular tiling graph T ∞ (i.e., the two-dimensional triangular grid) is the Archimedean tiling 3 6 introduced in  and . By a triangular grid graph G in this paper we mean a graph obtained from T ∞ as follows. Specify a number of triangles, called cells, in T ∞ . The edges of G are then all the edges surrounding the specified cells, while the vertices of G are the endpoints of the edges (defined by intersecting lines in T ∞ ). We say that the specified cells, along with any other cell whose all edges are from G, belong to G. Any triangular grid graph is 3-colorable, and thus it is word-representable . We consider non-3-colorable graphs obtained from triangular grid graphs by applying the operation of face subdivision which is defined in the sequel. ### A comprehensive introduction to the theory of word-representable graphs

Thus semi-transitive orientations generalize transitive orientations. A way to check if a given oriented graph G is semi-transitively ori- ented is as follows. First check that G is acyclic; if not, the orientation is not semi-transitive. Next, for a directed edge from a vertex x to a vertex y, consider each directed path P having at least three edges without repeated vertices from x to y, and check that the subgraph of G induced by P is transitive. If such non-transitive subgraph is found, the orientation is not semi-transitive. This procedure needs to be applied to each edge in G, and if no non-transitivity is discovered, G’s orientation is semi-transitive. ### Relation Prediction for Unseen Entities Using Entity Word Graphs

In this study, we proposed a method to learn en- tity representations using entity descriptions via graph structure. In the experiments, the perfor- mance of the proposed model showed a signiﬁ- cant improvement on the FB20K; furthermore, it outperforms the previous models on the FB15K. However, although the word order information (e.g., phrase) is an important clue for the re- lation prediction, our model disregards it when creating the Entity-Word graph. Thus, in fu- ture research, we plan to integrade our encoder with LSTM (Hochreiter and Schmidhuber, 1997) which can capture the word order information. ### Essentia: Mining Domain specific Paraphrases with Word Alignment Graphs

Paraphrases are important linguistic resources for a wide variety of NLP applications. Many techniques for automatic paraphrase mining from general corpora have been proposed. While these techniques are successful at dis- covering generic paraphrases, they often fail to identify domain-specific paraphrases (e.g., {“staff ”, “concierge”} in the hospitality do- main). This is because current techniques are often based on statistical methods, while domain-specific corpora are too small to fit sta- tistical methods. In this paper, we present an unsupervised graph-based technique to mine paraphrases from a small set of sentences that roughly share the same topic or intent. Our system, E SSENTIA , relies on word-alignment techniques to create a word-alignment graph that merges and organizes tokens from input sentences. The resulting graph is then used to generate candidate paraphrases. We demon- strate that our system obtains high quality paraphrases, as evaluated by crowd workers. We further show that the majority of the iden- tified paraphrases are domain-specific and thus complement existing paraphrase databases. ### Co-occurrence graphs for word sense disambiguation in the biomedical domain

Word Sense Disambiguation is a key step for many Natural Language Processing tasks (e.g. sum- marization, text classification, relation extraction) and presents a challenge to any system that aims to process documents from the biomedical domain. In this paper, we present a new graph- based unsupervised technique to address this problem. The knowledge base used in this work is a graph built with co-occurrence information from medical concepts found in scientific abstracts, and hence adapted to the specific domain. Unlike other unsupervised approaches based on static graphs such as UMLS, in this work the knowledge base takes the context of the ambiguous terms into account. Abstracts downloaded from PubMed are used for building the graph and disam- biguation is performed using the Personalized PageRank algorithm. Evaluation is carried out over two test datasets widely explored in the literature. Different parameters of the system are also evaluated to test robustness and scalability. Results show that the system is able to outperform state-of-the-art knowledge-based systems, obtaining more than 10% of accuracy improvement in some cases, while only requiring minimal external resources. ### Word Sense Induction & Disambiguation Using Hierarchical Random Graphs

Our future work focuses on using different feature types, e.g. dependency relations, second-order co- occurrences, named entities and others to construct our undirected graphs and then applying HRGs, in order to measure the impact of each feature type on the induced hierarchical structures within a WSD setting. Moreover, following the work in (Clauset et al., 2008), we are also working on using MCMC in order to sample more than one dendrogram at equi- librium, and then combine them to a consensus tree. This consensus tree might be able to express a larger amount of topological features of the initial undi- rected graph. ### The equational theories of representable residuated semigroups

It is easily checked that the above axioms are valid in representable al- gebras. We just note, in connection with the last two axioms, that the interpretation of reflexive residuated elements x must include the identity (they are reflexive) and they are transitive (x ; x ≤ x). ### Classes of representable disjoint NP-pairs

In the previous section we briefly explained how properties of propositional proof systems can be captured by disjoint NP -pairs that are suitably defined from these proof systems. Conversely, we now employ proof-theoretic methods to gain a more detailed understanding of the class of disjoint NP -pairs. For this we need to represent arbitrary disjoint NP -pairs in propositional proof systems. This can be done uniformly in theories of bounded arithmetic or non-uniformly in propositional proof systems. We will start with the uniform concept which was first considered by Razborov . ### p representable operators in Banach spaces

Let LE,F be the space of all bounded linear operators from E into F and BE* the unit ball of E*, the dual of E The completion of the injective tensor product of E and F is denoted by E F[r] ### Nondestructive Evaluation Of Corrosion Damage In Reinforced Concrete Aged Slab Specimen

The ability to repeat speech is impaired in most individuals with aphasia. Recent evidence suggests damage to area Spt (boundary of the parietal and temporal lobes at the Sylvian fissure) may cause the repetition difficulties commonly seen in aphasia. This study examined if such repetition impairments are specific to speech or reflect a more general repetition deficit, and determined how regional and network brain damage predict repetition impairments. Participants in the chronic phase of stroke (N=47) listened to a series of ten five-second melodies that consisted of six tones and repeated the melody (by humming) following its presentation. The participants’ audio samples were rated based on their similarity to the target melody, using a sentiment scale. The sentiment scale included the following ratings: strongly negative, negative, neutral, positive, and strongly positive. The audio samples were given one of these ratings based on their accuracy compared to the target melody. These scores were compared with the Western Aphasia Battery (WAB) repetition subscores to relate real word repetition to melody repetition. Melody repetition scores were also compared to nonword repetition by using a nonword word repetition task. A moderate association between melodic repetition and speech (real word and nonword) repetition was observed. Several connections were implicated as predicting poorer performance on the three behavioral tasks. A common shared 