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Counting Small Induced Subgraphs with Hereditary Properties [article]

Jacob Focke, Marc Roth
2022 arXiv   pre-print
We study the computational complexity of the problem #IndSub(Φ) of counting k-vertex induced subgraphs of a graph G that satisfy a graph property Φ.  ...  Our main result establishes an exhaustive and explicit classification for all hereditary properties, including tight conditional lower bounds under the Exponential Time Hypothesis (ETH): - If a hereditary  ...  In summary, together with Khot and Raman [24] , and Meeks [25] , we fully complete the complexity landscape for detection, approximate counting and exact counting induced subgraphs with hereditary properties  ... 
arXiv:2111.02277v2 fatcat:nitc2h2ykzcb5ffqiabmtkfehq

The Implicit Graph Conjecture is False [article]

Hamed Hatami, Pooya Hatami
2021 arXiv   pre-print
We refute this conjecture by establishing the existence of hereditary graph families with factorial speed of growth that require codes of length n^Ω(1).  ...  The Implicit Graph Conjecture states that, conversely, every hereditary graph family with at most factorial speed of growth admits an efficient implicit representation.  ...  A graph family F is called hereditary if it is closed under taking induced subgraphs.  ... 
arXiv:2111.13198v2 fatcat:3l2gfon2frgwfg76opn3cke7vm

Counting Small Induced Subgraphs Satisfying Monotone Properties

Marc Roth, Johannes Schmitt, Philip Wellnitz
2020 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)  
Sci. 15] and is part of the major line of research on counting small patterns in graphs.  ...  Given a graph property Φ, the problem #IndSub(Φ) asks, on input a graph G and a positive integer k, to compute the number #IndSub(Φ, k → G) of induced subgraphs of size k in G that satisfy Φ.  ...  Given any graph H, the property Φ of being H-free, that is, not containing H as an induced subgraph, is hereditary with Γ(Φ) = {H}.  ... 
doi:10.1109/focs46700.2020.00128 fatcat:5c4e6tpanrhjle537ksibs4m6q

Counting Small Induced Subgraphs Satisfying Monotone Properties [article]

Marc Roth and Johannes Schmitt and Philip Wellnitz
2020 arXiv   pre-print
Sci. 15] and is part of the major line of research on counting small patterns in graphs.  ...  Given a graph property Φ, the problem #IndSub(Φ) asks, on input a graph G and a positive integer k, to compute the number of induced subgraphs of size k in G that satisfy Φ.  ...  Given any graph H, the property Φ of being H-free, that is, not containing H as an induced subgraph, is hereditary with Γ(Φ) = {H}.  ... 
arXiv:2004.06595v1 fatcat:dhkmz5tw3bdipkdvr5645dcho4

Testing Hereditary Properties of Nonexpanding Bounded-Degree Graphs

Artur Czumaj, Asaf Shapira, Christian Sohler
2009 SIAM journal on computing (Print)  
As an application, our result implies that, for example, any hereditary property (e.g., k-colorability, H-freeness, etc.) is testable in the bounded degree graph model for planar graphs, graphs with bounded  ...  A graph family is hereditary if it is closed under vertex removal. Similarly, a graph property is hereditary if it is closed under vertex removal.  ...  an induced subgraph.  ... 
doi:10.1137/070681831 fatcat:qni3x5numvhi7kbnvdqarznziu

Counting Subgraphs in Somewhere Dense Graphs [article]

Marco Bressan, Leslie Ann Goldberg, Kitty Meeks, Marc Roth
2022 arXiv   pre-print
We study the problems of counting copies and induced copies of a small pattern graph H in a large host graph G.  ...  Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes 𝒢 as our central objects of study and establish the following crisp dichotomies  ...  For counting small induced subgraphs, i.e., for #IndSub(H → G), we will rely on a different transformation.  ... 
arXiv:2209.03402v1 fatcat:fezwi657sna6dnsfyx5r72eqcq

Largest Chordal and Interval Subgraphs Faster than $$2^n$$ 2 n

Ivan Bliznets, Fedor V. Fomin, Michał Pilipczuk, Yngve Villanger
2015 Algorithmica  
We prove that in a graph with n vertices, induced chordal and interval subgraphs with the maximum number of vertices can be found in time O(2 λn ) for some λ < 1.  ...  [10] have shown that for every hereditary class of graphs Π that have constant treewidth and are definable in counting monadic second-order logic (CMSO), the Maximum Induced Π -Subgraph problem can  ...  (Let us recall that a class of graphs is hereditary if it is closed under taking induced subgraphs).  ... 
doi:10.1007/s00453-015-0054-2 fatcat:xhgulpvdmza4tbtykpxbgvifma

The unlabelled speed of a hereditary graph property

József Balogh, Béla Bollobás, Michael Saks, Vera T. Sós
2009 Journal of combinatorial theory. Series B (Print)  
A property of graphs is a collection P of graphs closed under isomorphism; we call P hereditary if it is closed under taking induced subgraphs.  ...  Recent work on hereditary graph properties has shown that "large" and "small" labelled speeds of hereditary graph properties do jump.  ...  In what follows, by a "subgraph" we always mean an induced subgraph, so that a graph property is hereditary if it is closed under taking subgraphs.  ... 
doi:10.1016/j.jctb.2008.03.004 fatcat:o4plouncgffdfha5azetyw5bqa

Largest chordal and interval subgraphs faster than 2^n [article]

Ivan Bliznets, Fedor V. Fomin, Michał Pilipczuk, Yngve Villanger
2013 arXiv   pre-print
We prove that in an n-vertex graph, induced chordal and interval subgraphs with the maximum number of vertices can be found in time O(2^λ n) for some λ<1.  ...  As Π is hereditary, it may be described by a list of vertex-minimal forbidden induced subgraphs F Π . We need the following properties of F Π : Property (2).  ...  Property Time Since every hereditary class of graphs Π can be characterized by a (not necessarily finite) set of forbidden induced subgraphs, there is an equivalent formulation of the Maximum Induced Π-Subgraph  ... 
arXiv:1311.4055v1 fatcat:3h6sotkbybaxteymdhdoyccr4u

Graph parameters, Ramsey theory and the speed of hereditary properties [article]

Vadim Lozin
2018 arXiv   pre-print
The speed of a hereditary property P is the number P_n of n-vertex labelled graphs in P.  ...  In particular, we show that the speed of a hereditary property P is sub-factorial if and only if the neighbourhood diversity of graphs in P is bounded by a constant, and that the entropy of a hereditary  ...  Introduction A graph property is an infinite class of graphs closed under isomorphism. A property is hereditary if it is closed under taking induced subgraphs.  ... 
arXiv:1608.07727v2 fatcat:77ui7ca23jhrzcf2kxgj4exhde

On generalized perfect graphs: bounded degree and bounded edge perfection

Edward R. Scheinerman, Ann Trenk
1993 Discrete Applied Mathematics  
Whenever 9 is a hereditary property, it is natural to consider the minimal (with respect to the 5, "is an induced subgraph" partial ordering) graphs not in 9.  ...  A graph is perfect iff it does not contain C,,+r or C,,+t (with k> 1) as an induced subgraph.  ... 
doi:10.1016/0166-218x(93)90234-f fatcat:hcb6sb2wevdchpeyncnq5elt4a

Hereditarily extended properties, quasi-random graphs and not necessarily induced subgraphs

Mikl�s Simonovits, Vera T. S�s
1997 Combinatorica  
Here the properties are strongly connected to counting not necessarily induced subgraphs of a given type, while in a subsequent paper we shall investigate the properties connected with counting induced  ...  Given a subset X of V(G), e(X) denotes the number of edges of the subgraph induced by X, and G[X] denotes the subgraph of G spanned by X. Given two disjoint subsets X and Y in V(G),  ...  We shall call a property hereditary if it is assumed for all the sufficiently large induced subgraphs F h of our graph, but (only) with the same weaker error-term Here we shall consider properties P in  ... 
doi:10.1007/bf01195005 fatcat:riblmkxljrf4bn2tcj7ml7qoqe

On the Advice Complexity of Online Edge- and Node-Deletion Problems [chapter]

Peter Rossmanith
2018 Lecture Notes in Computer Science  
We consider only hereditary properties Π, for which optimal online algorithms exist and which can be characterized by a set of forbidden subgraphs F .  ...  For node-deletion problems we characterize the advice complexity exactly for all cases and for edge-deletion problems at least for the case of a single forbidden induced subgraph.  ...  One important case are hereditary properties Π, i.e., properties that are closed under taking induced subgraphs.  ... 
doi:10.1007/978-3-319-98355-4_26 fatcat:3hju5gjtcrcl5ffpedvf5nufne

Page 4877 of Mathematical Reviews Vol. , Issue 89I [page]

1989 Mathematical Reviews  
A graph is called distance-hereditary if for all connected induced subgraphs F of G, the distance between any given pair of vertices in F is the same as in G.  ...  Zbigniew Palka (PL-POZN) 05C Graph theory 89i:05248 89i:05245 05C80 60C05 Buckley, Fred (1-CUNY2); Palka, Zbigniew (PL-POZN) Random graphs with the distance-hereditary property Eighteenth Southeastern  ... 

Decomposing a graph into expanding subgraphs

Guy Moshkovitz, Asaf Shapira
2017 Random structures & algorithms (Print)  
author have recently shown that every graph with average degree d contains an n-vertex subgraph with average degree at least (1−o(1))d and vertex expansion 1/(log n) 1+o(1) .  ...  These graphs have a super-linear number of edges and nearly logarithmic girth, yet each of their subgraphs has (optimally) poor expansion properties.  ...  Edge density of hereditary families with small separators A family of graphs is said to be hereditary if it is closed under taking induced subgraphs.  ... 
doi:10.1002/rsa.20727 fatcat:kdm4y7jooveylfwbqhpw2x26tu
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