Generalization bounds for learning under graph-dependence: A survey

Zhang, Rui-Ray; Amini, Massih-Reza

doi:10.48550/arxiv.2203.13534

Cited by 2 publications

(2 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A tree-partition is a T-partition for some tree T. The tree-partition-width 3 of G, denoted by tpw(G), is the minimum width of a tree-partition of G. Thus tpw(G) = tpw 1 (G), which equals the minimum ∈ N 0 for which G is contained in T K for some tree T. Tree-partitions were independently introduced by Seese [69] and Halin [39], and have since been widely investigated [7,8,19,20,24,34,76,77]. Applications of tree-partitions include graph drawing [13,16,30,32,80], nonrepetitive graph colouring [2], clustered graph colouring [1], monadic second-order logic [51], network emulations [4,5,9,37], size Ramsey number [26,43], statistical learning theory [81], and the edge-Erdős-Pósa property [14,38,60]. Planar-partitions and other more general structures have also been studied [18,21,22,63,80].…”

Section: Partitionsmentioning

confidence: 99%

Product structure of graph classes with bounded treewidth

Campbell,

Clinch,

Distel

et al. 2023

Combinator. Probab. Comp.

View full text Add to dashboard Cite

We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$ , for every graph $G \in \mathcal{G}$ there is a graph $H$ with $\textrm{tw}(H) \leqslant c$ such that $G$ is isomorphic to a subgraph of $H \boxtimes K_{f(\textrm{tw}(G))}$ . We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth $3$ ; the class of $K_{s,t}$ -minor-free graphs has underlying treewidth $s$ (for $t \geqslant \max \{s,3\}$ ); and the class of $K_t$ -minor-free graphs has underlying treewidth $t-2$ . In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star.

show abstract

Section: Partitionsmentioning

confidence: 99%

Product structure of graph classes with bounded treewidth

Campbell,

Clinch,

Distel

et al. 2023

Combinator. Probab. Comp.

View full text Add to dashboard Cite

show abstract

“…Tree-partitions were independently introduced by Seese [68] and Halin [38], and have since been widely investigated [7,8,19,20,31,75,76]. Applications of tree-partitions include graph drawing [12,15,28,30,79], nonrepetitive graph colouring [2], clustered graph colouring [1], monadic second-order logic [50], network emulations [4,5,9,36], statistical learning theory [80], and the edge-Erdős-Pósa property [13,37,59]. Planar-partitions and other more general structures have also been studied [17,21,22,62,79].…”

Section: Partitionsmentioning

confidence: 99%

Product structure of graph classes with bounded treewidth

Campbell¹,

Clinch²,

Marc³

et al. 2022

Preprint

View full text Add to dashboard Cite

We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class G to be the minimum non-negative integer c such that, for some function f , for every graph G ∈ G there is a graphWe introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth 3; the class of K s,t -minor-free graphs has underlying treewidth s (for t max{s, 3}); and the class of K t -minor-free graphs has underlying treewidth t − 2. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no H subgraph has bounded underlying treewidth if and only if every component of H is a subdivided star, and that the class of graphs with no induced H subgraph has bounded underlying treewidth if and only if every component of H is a star.

show abstract

Generalization bounds for learning under graph-dependence: A survey

Cited by 2 publications

References 20 publications

Product structure of graph classes with bounded treewidth

Product structure of graph classes with bounded treewidth

Product structure of graph classes with bounded treewidth

Contact Info

Product

Resources

About