scholarly journals Cops and robbers on graphs with a set of forbidden induced subgraphs

2020 ◽  
Vol 839 ◽  
pp. 186-194
Author(s):  
Masood Masjoody ◽  
Ladislav Stacho
Keyword(s):  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Cops And Robbers

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

scholarly journals A complete classification of the complexity of the vertex 3-colourability problem for quadruples of induced 5-vertex prohibitions

2020 ◽  
Vol 22 (1) ◽  
pp. 38-47
Author(s):  
Dmitry S. Malyshev
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Complete Classification ◽  
Induced Subgraphs ◽  
Simple Graphs ◽  
Forbidden Induced Subgraphs ◽  
The Third ◽  
Hereditary Class ◽  
Complexity Status

The vertex 3-colourability problem for a given graph is to check whether it is possible to split the set of its vertices into three subsets of pairwise non-adjacent vertices or not. A hereditary class of graphs is a set of simple graphs closed under isomorphism and deletion of vertices; the set of its forbidden induced subgraphs defines every such a class. For all but three the quadruples of 5-vertex forbidden induced subgraphs, we know the complexity status of the vertex 3-colourability problem. Additionally, two of these three cases are polynomially equivalent; they also polynomially reduce to the third one. In this paper, we prove that the computational complexity of the considered problem in all of the three mentioned classes is polynomial. This result contributes to the algorithmic graph theory.

scholarly journals A Note on the Speed of Hereditary Graph Properties

10.37236/644 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vadim V. Lozin ◽  
Colin Mayhill ◽  
Victor Zamaraev
Keyword(s):  
Practical Importance ◽  
Bipartite Graphs ◽  
Hereditary Property ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Graph Properties ◽  
Graph Property ◽  
Vertex Set ◽  
Split Graphs ◽  
Hereditary Properties

For a graph property $X$, let $X_n$ be the number of graphs with vertex set $\{1,\ldots,n\}$ having property $X$, also known as the speed of $X$. A property $X$ is called factorial if $X$ is hereditary (i.e. closed under taking induced subgraphs) and $n^{c_1n}\le X_n\le n^{c_2n}$ for some positive constants $c_1$ and $c_2$. Hereditary properties with the speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored, although this family includes many properties of theoretical or practical importance, such as planar graphs or graphs of bounded vertex degree. To simplify the study of factorial properties, we propose the following conjecture: the speed of a hereditary property $X$ is factorial if and only if the fastest of the following three properties is factorial: bipartite graphs in $X$, co-bipartite graphs in $X$ and split graphs in $X$. In this note, we verify the conjecture for hereditary properties defined by forbidden induced subgraphs with at most 4 vertices.

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