Large induced trees in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>K</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:math>-free graphs

Fox, Jacob; Loh, Po‐Shen; Sudakov, Benny

doi:10.1016/j.jctb.2008.10.001

Cited by 13 publications

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References 16 publications

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“…tree [10] log n 4 log r connected, K r -free tree [12] √ n connected, triangle-free tree [12] log n 12 log log n planar, 3-connected path [6] n − m + c 4 max degree ≤ 3 c = # connected components forest [1] n ⌈(∆ + 1)/3⌉ max degree ∆ max degree 2 [13] 3n ∆ + 5/3 max degree ∆ outerplanar [19] 3n ∆ + 1 max degree ∆ planar [7] 3n 2m/n + 1 m ≥ 2n or connected and m ≥ n has at most one cycle. Equivalently, the pseudoforests can be formed from forests (acyclic undirected graphs) by adding at most one edge per connected component.…”

Section: Preliminariesmentioning

confidence: 99%

Planar Induced Subgraphs of Sparse Graphs

Borradaile

Eppstein

Zhu

2014

Lecture Notes in Computer Science

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We show that every graph has an induced pseudoforest of at least n − m/4.5 vertices, an induced partial 2-tree of at least n − m/5 vertices, and an induced planar subgraph of at least n−m/5.2174 vertices. These results are constructive, implying linear-time algorithms to find the respective induced subgraphs. We also show that the size of the largest K h -minorfree graph in a given graph can sometimes be at most n − m/6 + o(m).

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Section: Preliminariesmentioning

confidence: 99%

Planar Induced Subgraphs of Sparse Graphs

Borradaile

Eppstein

Zhu

2014

Lecture Notes in Computer Science

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“…The studied approaches for such problems include parameterized complexity (Yannakakis and Gavril, 1987; Jaffke et al., 2020b; Bergougnoux and Kanté, 2021) and integer programming (Melo and Ribeiro, 2022). Several works deal with computational complexity and graph theoretical properties (Erdös et al., 1986; Palka and Ruciński, 1986; Frieze and Jackson, 1987; Suen, 1992; Scott, 1997; Rautenbach, 2007; Matoušek and Šámal, 2008; Derhy and Picouleau, 2009; Fox et al., 2009; Pfender, 2010; Hertz et al., 2014; Dutta and Subramanian, 2023). Other problems related to obtaining induced trees including a specified subset of the vertices were also considered in the literature.…”

Section: Literature Reviewmentioning

confidence: 99%

MIP formulations for induced graph optimization problems: a tutorial

Melo

Ribeiro

2023

Int Trans Operational Res

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Given a graph G=false(V,Efalse)$G=(V,E)$ and a subset of its vertices V′⊆V$V^{\prime }\subseteq V$, the subgraph induced by V′$V^{\prime }$ in G is that with vertex set V′$V^{\prime }$ and edge set E′$E^{\prime }$ formed by all the edges in E linking two vertices in V′$V^{\prime }$. Mixed integer programming (MIP) approaches are among the most successful techniques for solving induced graph optimization problems, that is, those related to obtaining maximum or minimum (weighted or not) induced subgraphs with certain properties. In this tutorial, we provide a literature review of some of these problems. Furthermore, we illustrate the use of MIP formulations and techniques for solving combinatorial optimization problems involving induced graphs. We focus on compact formulations and those with an exponential number of constraints that can be effectively solved using branch‐and‐cut procedures. More specifically, we revisit applications of their use for problems of finding induced forests (which correspond to the complement of feedback vertex sets), trees, paths, as well as quasi‐clique partitionings.

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Rooted induced trees in triangle‐free graphs

Pfender

2009

Journal of Graph Theory

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For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. Further, for a vertex v ∈ V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of G that is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle-free graphs G (and vertices v ∈ V(G)) on n vertices is denoted by t 3 (n) (t * 3 (n)). Clearly, t(G, v) ≤ t(G) for all v ∈ V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that G is connected and triangle-free. We show thatand determine the unique extremal graphs. Thus, we get as corollary that t 3 (n) ≥ t * 3 (n) = 1 2 (1+ √ 8n−7) , improving a recent result by Fox, Loh and Sudakov. ᭧

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Large induced trees in Kr-free graphs

Cited by 13 publications

References 16 publications

Planar Induced Subgraphs of Sparse Graphs

Planar Induced Subgraphs of Sparse Graphs

MIP formulations for induced graph optimization problems: a tutorial

Rooted induced trees in triangle‐free graphs

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