Minimal Obstructions for Partial Representations of Interval Graphs

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension

On some simplicial elimination schemes for chordal graphs

We present here some results on particular elimination schemes for chordal graphs, namely we show that for any chordal graph we can construct in linear time a simplicial elimination scheme starting with a pending maximal clique attached via a minimal separator maximal (resp. minimal) under inclusion among all minimal separators

Universal graphs with a forbidden subtree

We show that the problem of the existence of universal graphs with specified forbidden subgraphs can be systematically reduced to certain critical cases by a simple pruning technique which simplifies the underlying structure of the forbidden graphs, viewed as trees of blocks. As an application, we characterize the trees T for which a universal countable T-free graph exists

A Survey of Community Search Over Big Graphs

With the rapid development of information technologies, various big graphs are prevalent in many real applications (e.g., social media and knowledge bases). An important component of these graphs is the network community. Essentially, a community is a group of vertices which are densely connected internally. Community retrieval can be used in many real applications, such as event organization, friend recommendation, and so on. Consequently, how to efficiently find high-quality communities from big graphs is an important research topic in the era of big data. Recently a large group of research works, called community search, have been proposed. They aim to provide efficient solutions for searching high-quality communities from large networks in real-time. Nevertheless, these works focus on different types of graphs and formulate communities in different manners, and thus it is desirable to have a comprehensive review of these works. In this survey, we conduct a thorough review of existing community search works. Moreover, we analyze and compare the quality of communities under their models, and the performance of different solutions. Furthermore, we point out new research directions. This survey does not only help researchers to have a better understanding of existing community search solutions, but also provides practitioners a better judgment on choosing the proper solutions

On Finding Lekkerkerker-Boland Subgraphs

Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a linear-time algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatrices of matrices that do not have the consecutive-ones property. We give a linear-time algorithm to find one in any matrix that does not have the consecutive-ones property.Comment: Submitted to WG 201

Optimizing Adiabatic Quantum Program Compilation using a Graph-Theoretic Framework

Adiabatic quantum computing has evolved in recent years from a theoretical field into an immensely practical area, a change partially sparked by D-Wave System's quantum annealing hardware. These multimillion-dollar quantum annealers offer the potential to solve optimization problems millions of times faster than classical heuristics, prompting researchers at Google, NASA and Lockheed Martin to study how these computers can be applied to complex real-world problems such as NASA rover missions. Unfortunately, compiling (embedding) an optimization problem into the annealing hardware is itself a difficult optimization problem and a major bottleneck currently preventing widespread adoption. Additionally, while finding a single embedding is difficult, no generalized method is known for tuning embeddings to use minimal hardware resources. To address these barriers, we introduce a graph-theoretic framework for developing structured embedding algorithms. Using this framework, we introduce a biclique virtual hardware layer to provide a simplified interface to the physical hardware. Additionally, we exploit bipartite structure in quantum programs using odd cycle transversal (OCT) decompositions. By coupling an OCT-based embedding algorithm with new, generalized reduction methods, we develop a new baseline for embedding a wide range of optimization problems into fault-free D-Wave annealing hardware. To encourage the reuse and extension of these techniques, we provide an implementation of the framework and embedding algorithms

A Characterization of Substar Graphs

The intersection graphs of stars in some tree are known as substar graphs. In this paper we give a characterization of substar graphs by the list of minimal forbidden induced subgraphs. This corrects a flaw in the main result of Chang, Jacobson, Monma and West (Subtree and substar intersection numbers, Discrete Appl. Math. 44, 205-220 (1993)) and this leads to a different list of minimal forbidden induced subgraphs.Comment: 7 page

The List Distinguishing Number Equals the Distinguishing Number for Interval Graphs

A \textit{distinguishing coloring} of a graph $G$ is a coloring of the vertices so that every nontrivial automorphism of $G$ maps some vertex to a vertex with a different color. The \textit{distinguishing number} of $G$ is the minimum $k$ such that $G$ has a distinguishing coloring where each vertex is assigned a color from $\{1,\ldots,k\}$. A \textit{list assignment} to $G$ is an assignment $L=\{L(v)\}_{v\in V(G)}$ of lists of colors to the vertices of $G$. A \textit{distinguishing $L$-coloring} of $G$ is a distinguishing coloring of $G$ where the color of each vertex $v$ comes from $L(v)$. The {\it list distinguishing number} of $G$ is the minimum $k$ such that every list assignment to $G$ in which $|L(v)|=k$ for all $v\in V(G)$ yields a distinguishing $L$-coloring of $G$. We prove that if $G$ is an interval graph, then its distinguishing number and list distinguishing number are equal.Comment: 11 page

Some results on triangle partitions

We show that there exist efficient algorithms for the triangle packing problem in colored permutation graphs, complete multipartite graphs, distance-hereditary graphs, k-modular permutation graphs and complements of k-partite graphs (when k is fixed). We show that there is an efficient algorithm for C_4-packing on bipartite permutation graphs and we show that C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite graphs that have a triangle partition

Confluent Drawings: Visualizing Non-planar Diagrams in a Planar Way

In this paper, we introduce a new approach for drawing diagrams that have applications in software visualization. Our approach is to use a technique we call confluent drawing for visualizing non-planar diagrams in a planar way. This approach allows us to draw, in a crossing-free manner, graphs--such as software interaction diagrams--that would normally have many crossings. The main idea of this approach is quite simple: we allow groups of edges to be merged together and drawn as "tracks" (similar to train tracks). Producing such confluent diagrams automatically from a graph with many crossings is quite challenging, however, so we offer two heuristic algorithms to test if a non-planar graph can be drawn efficiently in a confluent way. In addition, we identify several large classes of graphs that can be completely categorized as being either confluently drawable or confluently non-drawable.Comment: 10 pages, 18 figure