On the 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 y. The motivation to study representable graphs came from algebra, but this subject is interesting from graph theoretical, computer science, and combinatorics on words points of view. In this paper, we prove that for n greater than 3, the line graph of an n-wheel is non-representable. This not only provides a new construction of non-representable 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 n greater than 4, the line graph of the complete graph is also non-representable. We then use these facts to prove that given a graph G which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of G k-times is guaranteed to be non-representable for k greater than 3.Comment: 10 pages, 5 figure

New results on word-representable graphs

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 $(x,y)\in E$ for each $x\neq y$. The set of word-representable graphs generalizes several important and well-studied graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3, etc. By answering an open question from [M. Halldorsson, S. Kitaev and A. Pyatkin, Alternation graphs, Lect. Notes Comput. Sci. 6986 (2011) 191--202. Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2011, Tepla Monastery, Czech Republic, June 21-24, 2011.], in the present paper we show that not all graphs of vertex degree at most 4 are word-representable. Combining this result with some previously known facts, we derive that the number of $n$-vertex word-representable graphs is $2^{\frac{n^2}{3}+o(n^2)}$

Structural Analysis of Boolean Equation Systems

We analyse the problem of solving Boolean equation systems through the use of structure graphs. The latter are obtained through an elegant set of Plotkin-style deduction rules. Our main contribution is that we show that equation systems with bisimilar structure graphs have the same solution. We show that our work conservatively extends earlier work, conducted by Keiren and Willemse, in which dependency graphs were used to analyse a subclass of Boolean equation systems, viz., equation systems in standard recursive form. We illustrate our approach by a small example, demonstrating the effect of simplifying an equation system through minimisation of its structure graph

On the number of types in sparse graphs

We prove that for every class of graphs $\mathcal{C}$ which is nowhere dense, as defined by Nesetril and Ossona de Mendez, and for every first order formula $\phi(\bar x,\bar y)$, whenever one draws a graph $G\in \mathcal{C}$ and a subset of its nodes $A$, the number of subsets of $A^{|\bar y|}$ which are of the form $\{\bar v\in A^{|\bar y|}\, \colon\, G\models\phi(\bar u,\bar v)\}$ for some valuation $\bar u$ of $\bar x$ in $G$ is bounded by $\mathcal{O}(|A|^{|\bar x|+\epsilon})$, for every $\epsilon>0$. This provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes. We also give two new proofs of upper bounds on quantities in nowhere dense classes which are relevant for their logical treatment. Firstly, we provide a new proof of the fact that nowhere dense classes are uniformly quasi-wide, implying explicit, polynomial upper bounds on the functions relating the two notions. Secondly, we give a new combinatorial proof of the result of Adler and Adler stating that every nowhere dense class of graphs is stable. In contrast to the previous proofs of the above results, our proofs are completely finitistic and constructive, and yield explicit and computable upper bounds on quantities related to uniform quasi-wideness (margins) and stability (ladder indices)

On word-representability of polyomino triangulations

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 $(x,y)$ is an edge in $E$. Some graphs are word-representable, others are not. It is known that a graph is word-representable if and only if it accepts a so-called semi-transitive orientation. The main result of this paper is showing that a triangulation of any convex polyomino is word-representable if and only if it is 3-colorable. We demonstrate that this statement is not true for an arbitrary polyomino. We also show that the graph obtained by replacing each $4$-cycle in a polyomino by the complete graph $K_4$ is word-representable. We employ semi-transitive orientations to obtain our results

On graphs with representation number 3

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 $(x,y)$ is an edge in $E$. A graph is word-representable if and only if it is $k$-word-representable for some $k$, that is, if there exists a word containing $k$ copies of each letter that represents the graph. Also, being $k$-word-representable implies being $(k+1)$-word-representable. The minimum $k$ such that a word-representable graph is $k$-word-representable, is called graph's representation number. Graphs with representation number 1 are complete graphs, while graphs with representation number 2 are circle graphs. The only fact known before this paper on the class of graphs with representation number 3, denoted by $\mathcal{R}_3$, is that the Petersen graph and triangular prism belong to this class. In this paper, we show that any prism belongs to $\mathcal{R}_3$, and that two particular operations of extending graphs preserve the property of being in $\mathcal{R}_3$. Further, we show that $\mathcal{R}_3$ is not included in a class of $c$-colorable graphs for a constant $c$. To this end, we extend three known results related to operations on graphs. We also show that ladder graphs used in the study of prisms are $2$-word-representable, and thus each ladder graph is a circle graph. Finally, we discuss $k$-word-representing comparability graphs via consideration of crown graphs, where we state some problems for further research

Semi-Transitive Orientations and Word-Representable Graphs

A graph $G=(V,E)$ is a \emph{word-representable graph} if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of $G$ is the minimum $k$ such that $G$ is a representable by a word, where each letter occurs $k$ times; such a $k$ exists for any word-representable graph. We show that the representation number of a word-representable graph on $n$ vertices is at most $2n$, while there exist graphs for which it is $n/2$.Comment: arXiv admin note: text overlap with arXiv:0810.031

The Complexity of Reasoning with FODD and GFODD

Recent work introduced Generalized First Order Decision Diagrams (GFODD) as a knowledge representation that is useful in mechanizing decision theoretic planning in relational domains. GFODDs generalize function-free first order logic and include numerical values and numerical generalizations of existential and universal quantification. Previous work presented heuristic inference algorithms for GFODDs and implemented these heuristics in systems for decision theoretic planning. In this paper, we study the complexity of the computational problems addressed by such implementations. In particular, we study the evaluation problem, the satisfiability problem, and the equivalence problem for GFODDs under the assumption that the size of the intended model is given with the problem, a restriction that guarantees decidability. Our results provide a complete characterization placing these problems within the polynomial hierarchy. The same characterization applies to the corresponding restriction of problems in first order logic, giving an interesting new avenue for efficient inference when the number of objects is bounded. Our results show that for $\Sigma_k$ formulas, and for corresponding GFODDs, evaluation and satisfiability are $\Sigma_k^p$ complete, and equivalence is $\Pi_{k+1}^p$ complete. For $\Pi_k$ formulas evaluation is $\Pi_k^p$ complete, satisfiability is one level higher and is $\Sigma_{k+1}^p$ complete, and equivalence is $\Pi_{k+1}^p$ complete.Comment: A short version of this paper appears in AAAI 2014. Version 2 includes a reorganization and some expanded proof