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.

scholarly journals Probe split graphs

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Van Bang Le ◽  
H. N. Ridder
Keyword(s):  
Undirected Graph ◽  
Recognition Algorithm ◽  
Not Given ◽  
Induced Subgraphs ◽  
Split Graph ◽  
Forbidden Induced Subgraphs ◽  
Recognition Algorithms ◽  
Vertex Set ◽  
International Audience ◽  
Split Graphs

Graphs and Algorithms International audience An undirected graph G=(V,E) is a probe split graph if its vertex set can be partitioned into two sets, N (non-probes) and P (probes) where N is independent and there exists E' ⊆ N× N such that G'=(V,E∪ E') is a split graph. Recently Chang et al. gave an O(V4(V+E)) time recognition algorithm for probe split graphs. In this article we give O(V2+VE) time recognition algorithms and characterisations by forbidden induced subgraphs both for the case when the partition into probes and non-probes is given, and when it is not given.

scholarly journals The Edit Distance Function and Symmetrization

10.37236/2262 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Ryan R. Martin
Keyword(s):  
Distance Function ◽  
Edit Distance ◽  
Symmetric Difference ◽  
Hereditary Property ◽  
Vertex Set ◽  
Hereditary Properties

The edit distance between two graphs on the same labeled vertex set is the size of the symmetric difference of the edge sets.  The distance between a graph, G, and a hereditary property, ℋ, is the minimum of the distance between G and each G'∈ℋ.  The edit distance function of ℋ is a function of p∈[0,1] and is the limit of the maximum normalized distance between a graph of density p and ℋ.This paper utilizes a method due to Sidorenko [Combinatorica 13(1), pp. 109-120], called "symmetrization", for computing the edit distance function of various hereditary properties.  For any graph H, Forb(H) denotes the property of not having an induced copy of H.  This paper gives some results regarding estimation of the function for an arbitrary hereditary property. This paper also gives the edit distance function for Forb(H), where H is a cycle on 9 or fewer vertices.

scholarly journals Efficient Testing of Bipartite Graphs for Forbidden Induced Subgraphs

2007 ◽  
Vol 37 (3) ◽  
pp. 959-976 ◽  
Author(s):  
Noga Alon ◽  
Eldar Fischer ◽  
Ilan Newman
Keyword(s):  
Bipartite Graphs ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs

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 Clique cycle transversals in graphs with few P₄'s

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Raquel Bravo ◽  
Sulamita Klein ◽  
Loana Tito Nogueira ◽  
Fábio Protti
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Finite Family ◽  
Feedback Vertex Set ◽  
Induced Subgraphs ◽  
Complete Subgraph ◽  
Forbidden Induced Subgraphs ◽  
Vertex Set ◽  
International Audience ◽  
Acyclic Subgraph

Graph Theory International audience A graph is extended P4-laden if each of its induced subgraphs with at most six vertices that contains more than two induced P4's is 2K2,C4-free. A cycle transversal (or feedback vertex set) of a graph G is a subset T ⊆ V (G) such that T ∩ V (C) 6= ∅ for every cycle C of G; if, in addition, T is a clique, then T is a clique cycle transversal (cct). Finding a cct in a graph G is equivalent to partitioning V (G) into subsets C and F such that C induces a complete subgraph and F an acyclic subgraph. This work considers the problem of characterizing extended P4-laden graphs admitting a cct. We characterize such graphs by means of a finite family of forbidden induced subgraphs, and present a linear-time algorithm to recognize them.

scholarly journals Finding Balance: Split Graphs and Related Classes

10.37236/7091 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Karen L. Collins ◽  
Ann N. Trenk
Keyword(s):  
Bipartite Graphs ◽  
Stable Set ◽  
Minimal Set ◽  
Split Graph ◽  
Vertex Set ◽  
Split Graphs

A graph is a split graph if its vertex set can be partitioned into a clique and a stable set. A split graph is unbalanced if there exist two such partitions that are distinct. Cheng, Collins and Trenk (2016), discovered the following interesting counting fact: unlabeled, unbalanced split graphs on $n$ vertices can be placed into a bijection with all unlabeled split graphs on $n-1$ or fewer vertices. In this paper we translate these concepts and the theorem to different combinatorial settings: minimal set covers, bipartite graphs with a distinguished block and posets of height one.

scholarly journals Forbidden Induced Subgraphs of Double-split Graphs

2012 ◽  
Vol 26 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Boris Alexeev ◽  
Alexandra Fradkin ◽  
Ilhee Kim
Keyword(s):  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Split Graphs

scholarly journals Partitioning the vertex set of a bipartite graph into complete bipartite subgraphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Oleg Duginov
Keyword(s):  
Bipartite Graphs ◽  
Partition Problem ◽  
Induced Subgraphs ◽  
Permutation Graphs ◽  
Vertex Partition ◽  
Vertex Set ◽  
International Audience ◽  
Bipartite Permutation Graphs ◽  
Np Complete ◽  
Polynomially Solvable

Graph Theory International audience Given a graph and a positive integer k, the biclique vertex-partition problem asks whether the vertex set of the graph can be partitioned into at most k bicliques (connected complete bipartite subgraphs). It is known that this problem is NP-complete for bipartite graphs. In this paper we investigate the computational complexity of this problem in special subclasses of bipartite graphs. We prove that the biclique vertex-partition problem is polynomially solvable for bipartite permutation graphs, bipartite distance-hereditary graphs and remains NP-complete for perfect elimination bipartite graphs and bipartite graphs containing no 4-cycles as induced subgraphs.

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