scholarly journals Minimal Separators in Graph Classes Defined by Small Forbidden Induced Subgraphs

2019 ◽  
pp. 379-391
Author(s):  
Martin Milanič ◽  
Nevena Pivač
Keyword(s):  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Graph Classes

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 Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem

10.37236/9428 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Martin Milanič ◽  
Nevena Pivač
Keyword(s):  
Cartesian Product ◽  
Sufficient Conditions ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Dichotomy Theorem ◽  
Bounded Number ◽  
Graph Classes ◽  
The Family ◽  
Graph Class ◽  
Hereditary Graph Class

A graph class is said to be tame if graphs in the class have a polynomially bounded number of minimal separators. Tame graph classes have good algorithmic properties, which follow, for example, from an algorithmic metatheorem of Fomin, Todinca, and Villanger from 2015. We show that a hereditary graph class $\mathcal{G}$ is tame if and only if the subclass consisting of graphs in $\mathcal{G}$ without clique cutsets is tame. This result and Ramsey's theorem lead to several types of sufficient conditions for a graph class to be tame. In particular, we show that any hereditary class of graphs of bounded clique cover number that excludes some complete prism is tame, where a complete prism is the Cartesian product of a complete graph with a $K_2$. We apply these results, combined with constructions of graphs with exponentially many minimal separators, to develop a dichotomy theorem separating tame from non-tame graph classes within the family of graph classes defined by sets of forbidden induced subgraphs with at most four vertices.

scholarly journals Clique-Width of Graph Classes Defined by Two Forbidden Induced Subgraphs

2015 ◽  
Vol 59 (5) ◽  
pp. 650-666 ◽  
Author(s):  
Konrad K. Dabrowski ◽  
Daniël Paulusma
Keyword(s):  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Graph Classes

scholarly journals Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs

2021 ◽  
Vol 295 ◽  
pp. 57-69
Author(s):  
A. Atminas ◽  
R. Brignall ◽  
V. Lozin ◽  
J. Stacho
Keyword(s):  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs

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 Forbidden induced subgraphs for star-free graphs

2011 ◽  
Vol 311 (21) ◽  
pp. 2475-2484 ◽  
Author(s):  
Jun Fujisawa ◽  
Katsuhiro Ota ◽  
Kenta Ozeki ◽  
Gabriel Sueiro
Keyword(s):  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs
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