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.

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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**.

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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.

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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.

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Related work. The notion of directed **word**-**representable** **graphs** was in- troduced in [13] 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 [12], numerous properties of **word**-**representable** **graphs** were derived and several types of **word**-**representable** and **non**-**word**-**representable** **graphs** pinpointed. Some open questions from [12] were resolved recently in [7], including the representability of the Petersen graph.

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The notion of a **word**-**representable** graph first appears in [4]. 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

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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.

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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.

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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.

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The triangular tiling graph T ∞ (i.e., the two-dimensional triangular grid) is the Archimedean tiling 3 6 introduced in [13] and [4]. 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** [7]. We consider **non**-3-colorable **graphs** obtained from triangular grid **graphs** by applying the operation of face subdivision which is defined in the sequel.

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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.

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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.

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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.

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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.

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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).

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 [24].

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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]

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

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Rationale: This area of instruction provides content for knowledge and skills required in the technology-based workplace. The demand will continue to expand for individuals to use computer hardware and software to create documents, gather information, and solve problems. This class is vital for students planning to enter the workforce or postsecondary education. Course Description: This course is designed to help students master beginning and advanced skills in the areas of **word** processing, database management, spreadsheet applications, desktop publishing, multimedia, Internet usage, and integrated software applications.

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