scholarly journals Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition

10.37236/6431 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Maryam Bahrani ◽  
Jérémie Lumbroso
Keyword(s):  
Asymptotic Analysis ◽  
Forbidden Subgraph ◽  
Decomposition Tree ◽  
Random Generation ◽  
Split Decomposition ◽  
Induced Subgraphs ◽  
Graph Classes ◽  
Cactus Graphs ◽  
Future Work ◽  
Natural Way

Forbidden characterizations may sometimes be the most natural way to describe families of graphs, and yet these characterizations are usually very hard to exploit for enumerative purposes.By building on the work of Gioan and Paul (2012) and Chauve et al.(2014), we show a methodology by which we constrain a split-decomposition tree to avoid certain patterns, thereby avoiding the corresponding induced subgraphs in the original graph.We thus provide the grammars and full enumeration for a wide set of graph classes: ptolemaic, block, and variants of cactus graphs (2,3-cacti, 3-cacti and 4-cacti). In certain cases, no enumeration was known (ptolemaic, 4-cacti); in other cases, although the enumerations were known, an abundant potential is unlocked by the grammars we provide (in terms of asymptotic analysis, random generation, and parameter analyses, etc.).We believe this methodology here shows its potential; the natural next step to develop its reach would be to study split-decomposition trees which contain certain prime nodes. This will be the object of future work. 

scholarly journals Matrix Partitions with Finitely Many Obstructions

10.37236/976 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Tomás Feder ◽  
Pavol Hell ◽  
Wing Xie
Keyword(s):  
Symmetric Matrix ◽  
Forbidden Subgraph ◽  
Input Graph ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Forbidden Induced Subgraphs ◽  
Extra Effort ◽  
Matrix Partition ◽  
Finite Set ◽  
Forbidden Induced Subgraph

Each $m$ by $m$ symmetric matrix $M$ over $0, 1, *$, defines a partition problem, in which an input graph $G$ is to be partitioned into $m$ parts with adjacencies governed by $M$, in the sense that two distinct vertices in (possibly equal) parts $i$ and $j$ are adjacent if $M(i,j)=1$, and nonadjacent if $M(i,j)=0$. (The entry $*$ implies no restriction.) We ask which matrix partition problems admit a characterization by a finite set of forbidden induced subgraphs. We prove that matrices containing a certain two by two diagonal submatrix $S$ never have such characterizations. We then develop a recursive technique that allows us (with some extra effort) to verify that matrices without $S$ of size five or less always have a finite forbidden induced subgraph characterization. However, we exhibit a six by six matrix without $S$ which cannot be characterized by finitely many induced subgraphs. We also explore the connection between finite forbidden subgraph characterizations and related questions on the descriptive and computational complexity of matrix partition problems.

scholarly journals Smaller Extended Formulations for Spanning Tree Polytopes in Minor-closed Classes and Beyond

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Manuel Aprile ◽  
Samuel Fiorini ◽  
Tony Huynh ◽  
Gwenaël Joret ◽  
David R. Wood
Keyword(s):  
Spanning Tree ◽  
Short Proof ◽  
Induced Subgraphs ◽  
Direct Construction ◽  
Extension Complexity ◽  
Graph Classes ◽  
Closed Class ◽  
Vertex Graph ◽  
Graph Class ◽  
Strongly Sublinear

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph class closed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$ have balanced separators of size $O(n^\beta)$, then the extension complexity of the spanning tree polytope of every connected $n$-vertex graph in $\mathcal{G}$ is $O(n^{1+\beta})$. We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the $O(n)$ bound for planar graphs due to Williams (2002).

scholarly journals Upper paired domination versus upper domination

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Hadi Alizadeh ◽  
Didem Gözüpek
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Graph Classes ◽  
Paired Domination Number ◽  
Upper Domination Number ◽  
Paired Domination ◽  
Cactus Graphs ◽  
Open Question

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.

scholarly journals Forbidden Subgraphs of Power Graphs

10.37236/9961 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Pallabi Manna ◽  
Peter J. Cameron ◽  
Ranjit Mehatari
Keyword(s):  
Nilpotent Groups ◽  
Chordal Graphs ◽  
Forbidden Subgraphs ◽  
Induced Subgraphs ◽  
Open Problems ◽  
Split Graph ◽  
Forbidden Induced Subgraphs ◽  
Power Graph ◽  
Graph Classes ◽  
Split Graphs

The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.

scholarly journals Survey on Interpretable Semantic Textual Similarity, and its Applications

2021 ◽  
Vol 10 (3) ◽  
pp. 14-18
Author(s):  
Abdo Ababor Abafogi
Keyword(s):  
Language Processing ◽  
Semantic Representation ◽  
Hybrid Approach ◽  
Machine Learning Approach ◽  
Paper Review ◽  
Future Direction ◽  
Rule Based Approach ◽  
Future Work ◽  
Natural Way ◽  
Semantic Textual Similarity

Both semantic representation and related natural language processing(NLP) tasks has become more popular due to the introduction of distributional semantics. Semantic textual similarity (STS)is one of a task in NLP, it determinesthe similarity based onthe meanings of two shorttexts (sentences). Interpretable STS is the way of giving explanation to semantic similarity between short texts. Giving interpretation is indeedpossible tohuman, but, constructing computational modelsthat explain as human level is challenging. The interpretable STS task give output in natural way with a continuous value on the scale from [0, 5] that represents the strength of semantic relation between pair sentences, where 0 is no similarity and 5 is complete similarity. This paper review all available methods were used in interpretable STS computation, classify them, specifyan existing limitations, and finally give directions for future work. This paper is organized the survey into nine sections as follows: firstly introduction at glance, then chunking techniques and available tools, the next one is rule based approach, the fourth section focus on machine learning approach, after that about works done via neural network, and the finally hybrid approach concerned. Application of interpretable STS, conclusion and future direction is also part of this paper.

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