scholarly journals Smaller Extended Formulations for Spanning Tree Polytopes in Minor-closed Classes and Beyond

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Manuel Aprile ◽  
Samuel Fiorini ◽  
Tony Huynh ◽  
Gwenaël Joret ◽  
David R. Wood
Keyword(s):  
Spanning Tree ◽  
Short Proof ◽  
Induced Subgraphs ◽  
Direct Construction ◽  
Extension Complexity ◽  
Graph Classes ◽  
Closed Class ◽  
Vertex Graph ◽  
Graph Class ◽  
Strongly Sublinear

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph class closed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$ have balanced separators of size $O(n^\beta)$, then the extension complexity of the spanning tree polytope of every connected $n$-vertex graph in $\mathcal{G}$ is $O(n^{1+\beta})$. We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the $O(n)$ bound for planar graphs due to Williams (2002).

scholarly journals Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem

10.37236/9428 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Martin Milanič ◽  
Nevena Pivač
Keyword(s):  
Cartesian Product ◽  
Sufficient Conditions ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Dichotomy Theorem ◽  
Bounded Number ◽  
Graph Classes ◽  
The Family ◽  
Graph Class ◽  
Hereditary Graph Class

A graph class is said to be tame if graphs in the class have a polynomially bounded number of minimal separators. Tame graph classes have good algorithmic properties, which follow, for example, from an algorithmic metatheorem of Fomin, Todinca, and Villanger from 2015. We show that a hereditary graph class $\mathcal{G}$ is tame if and only if the subclass consisting of graphs in $\mathcal{G}$ without clique cutsets is tame. This result and Ramsey's theorem lead to several types of sufficient conditions for a graph class to be tame. In particular, we show that any hereditary class of graphs of bounded clique cover number that excludes some complete prism is tame, where a complete prism is the Cartesian product of a complete graph with a $K_2$. We apply these results, combined with constructions of graphs with exponentially many minimal separators, to develop a dichotomy theorem separating tame from non-tame graph classes within the family of graph classes defined by sets of forbidden induced subgraphs with at most four vertices.

scholarly journals Expanders Are Universal for the Class of All Spanning Trees

2013 ◽  
Vol 22 (2) ◽  
pp. 253-281 ◽  
Author(s):  
DANIEL JOHANNSEN ◽  
MICHAEL KRIVELEVICH ◽  
WOJCIECH SAMOTIJ
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Spanning Trees ◽  
Clique Number ◽  
Interesting Consequence ◽  
Random Regular Graph ◽  
Vertex Graph ◽  
Graph Class ◽  
Universal Graphs ◽  
Natural Expansion

A graph is calleduniversalfor a given graph class(or, equivalently,-universal) if it contains a copy of every graph inas a subgraph. The construction of sparse universal graphs for various classeshas received a considerable amount of attention. There is particular interest in tight-universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from. Arguably, the most studied case is that whenis some class of trees. In this work, we are interested in(n,Δ), the class of alln-vertex trees with maximum degree at most Δ. We show that everyn-vertex graph satisfying certain natural expansion properties is(n,Δ)-universal. Our methods also apply to the case when Δ is some function ofn. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constantcsuch that the random graphG(n,cn−1/3log2n) is asymptotically almost surely (a.a.s.) universal for(n,O(1)). Moreover, a corresponding result holds for the random regular graph of degreecn2/3log2n. Another interesting consequence is the existence of locally sparsen-vertex(n,Δ)-universal graphs. For example, we show that one can (randomly) constructn-vertex(n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose(n,Δ)-universal graphs with merelyO(n)edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly(n,Δ)-universal in the context of the Maker–Breaker tree-universality game.

Sizes of Induced Subgraphs of Ramsey Graphs

2009 ◽  
Vol 18 (4) ◽  
pp. 459-476 ◽  
Author(s):  
NOGA ALON ◽  
JÓZSEF BALOGH ◽  
ALEXANDR KOSTOCHKA ◽  
WOJCIECH SAMOTIJ
Keyword(s):  
Lower Bound ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Vertex Graph ◽  
Ramsey Graphs

An n-vertex graph G is c-Ramsey if it contains neither a complete nor an empty induced subgraph of size greater than c log n. Erdős, Faudree and Sós conjectured that every c-Ramsey graph with n vertices contains Ω(n5/2) induced subgraphs, any two of which differ either in the number of vertices or in the number of edges, i.e., the number of distinct pairs (|V(H)|, |E(H)|), as H ranges over all induced subgraphs of G, is Ω(n5/2). We prove an Ω(n2.3693) lower bound.

scholarly journals Low-congestion shortcut and graph parameters

2021 ◽  
Author(s):  
Naoki Kitamura ◽  
Hirotaka Kitagawa ◽  
Yota Otachi ◽  
Taisuke Izumi
Keyword(s):  
Lower Bound ◽  
Minimum Spanning Tree ◽  
Hard Core ◽  
Construction Technique ◽  
Graph Problems ◽  
Graph Classes ◽  
Graph Class ◽  
Network Topologies ◽  
Graph Parameters

AbstractDistributed graph algorithms in the standard CONGEST model often exhibit the time-complexity lower bound of $${\tilde{\Omega }}(\sqrt{n} + D)$$ Ω ~ ( n + D ) rounds for several global problems, where n denotes the number of nodes and D the diameter of the input graph. Because such a lower bound is derived from special “hard-core” instances, it does not necessarily apply to specific popular graph classes such as planar graphs. The concept of low-congestion shortcuts was initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. In particular, given a graph class $${\mathcal {C}}$$ C , an f-round algorithm for constructing shortcuts of quality q for any instance in $${\mathcal {C}}$$ C results in $${\tilde{O}}(q + f)$$ O ~ ( q + f ) -round algorithms for solving several fundamental graph problems such as minimum spanning tree and minimum cut, for $${\mathcal {C}}$$ C . The main interest on this line is to identify the graph classes allowing the shortcuts that are efficient in the sense of breaking $${\tilde{O}}(\sqrt{n}+D)$$ O ~ ( n + D ) -round general lower bounds. In this study, we consider the relationship between the quality of low-congestion shortcuts and the following four major graph parameters: doubling dimension, chordality, diameter, and clique-width. The key ingredient of the upper-bound side is a novel shortcut construction technique known as short-hop extension, which might be of independent interest.

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 A Note on Random Minimum Length Spanning Trees

10.37236/1519 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Alan Frieze ◽  
Miklós Ruszinkó ◽  
Lubos Thoma
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Minimum Length ◽  
Edge Connectivity ◽  
Vertex Graph ◽  
Expected Length ◽  
Independent Edge

Consider a connected $r$-regular $n$-vertex graph $G$ with random independent edge lengths, each uniformly distributed on $[0,1]$. Let $mst(G)$ be the expected length of a minimum spanning tree. We show in this paper that if $G$ is sufficiently highly edge connected then the expected length of a minimum spanning tree is $\sim {n\over r}\zeta(3)$. If we omit the edge connectivity condition, then it is at most $\sim {n\over r}(\zeta(3)+1)$.

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