Related papers: Pebble Game Algorithms and Sparse Graphs
A hypergraph $G=(V,E)$ is $(k,\ell)$-sparse if no subset $V'\subset V$ spans more than $k|V'|-\ell$ hyperedges. We characterize $(k,\ell)$-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend…
We describe a new algorithm, the $(k,\ell)$-pebble game with colors, and use it obtain a characterization of the family of $(k,\ell)$-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs.…
Graph pebbling is a game played on a connected graph G. A player purchases pebbles at a dollar a piece, and hands them to an adversary who distributes them among the vertices of G (called a configuration) and chooses a target vertex r. The…
The family of $(k,\ell)$-sparse graphs, introduced by Lorea, plays a central role in combinatorial optimization and has a wide range of applications, particularly in rigidity theory. A key algorithmic problem is to decide whether a given…
A multigraph $G = (V, E)$ is $(k, \ell)$-sparse if every subset $X \subseteq V$ induces at most $\max\{k|X| - \ell, 0\}$ edges. Finding a maximum-size $(k, \ell)$-sparse subgraph is a classical problem in rigidity theory and combinatorial…
A pebbling move on a weighted graph removes some pebbles at a vertex and adds one pebble at an adjacent vertex. The number of pebbles removed is the weight of the edge connecting the vertices. A vertex is reachable from a pebble…
Graph Pebbling is a well-studied single-player game on graphs. We introduce the game of Blocking Pebbles which adapts Graph Pebbling into a two-player strategy game in order to examine it within the context of Combinatorial Game Theory.…
A set of vertices is $k$-sparse if it induces a graph with a maximum degree of at most $k$. In this missive, we consider the order of the largest $k$-sparse set in a triangle-free graph of fixed order. We show, for example, that every…
We expand the theory of pebbling to graphs with weighted edges. In a weighted pebbling game, one player distributes a set amount of weight on the edges of a graph and his opponent chooses a target vertex and places a configuration of…
A set of vertices in a graph is agile if, however we partition the set into two parts, we can always find two vertex-disjoint connected subgraphs where one covers the first and the other the second part. We present a characterization for…
The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$…
We say that a graph G is $(k,\ell)$-stable if removing $k$ vertices from it reduces its independence number by at most $\ell$. We say that G is tight $(k,\ell)$-stable if it is $(k,\ell)$-stable and its independence number equals…
The following measure of sparsity of multigraphs refining the maximum average degree: For $a>0$ and an arbitrary real $b$, a multigraph $H$ is \emph{$(a,b)$-sparse} if it is loopless and for every $A\subseteq V(H)$ with $|A|\geq 2$, the…
Graph pebbling is a game played on graphs with pebbles on their vertices. A pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. The pebbling number $\pi(G)$ is the smallest $t$ so that from any…
We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph has no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular…
We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph $G\sim {\mathcal G}(n,p)$ in order to find a subgraph which…
The simple graphs $G=(V,E)$ that satisfy $|E'|\leq 2|V'|-l$ for any subgraph (and for $l=1,2,3$) are the $(2,l)$-sparse graphs. Those that also satisfy $|E|=2|V|-l$ are the $(2,l)$-tight graphs. These can be characterised by their…
Suppose $G$ is a undirected simple graph. A $k-$subset of edges in $G$ without common vertices is called a $k-$matching and the number of such subsets is denoted by $p(G,k)$. The aim of this paper is to present exact formulas for $p(G,3)$,…
A graph $G$ is said to be a $(k,\ell)$-graph if its vertex set can be partitioned into $k$ independent sets and $\ell$ cliques. It is well established that the recognition problem for $(k,\ell)$-graphs is NP-complete whenever $k \geq 3$ or…
For integers $k>\ell\ge0$, a graph $G$ is $(k,\ell)$-stable if $\alpha(G-S)\geq \alpha(G)-\ell$ for every $S\subseteq V(G)$ with $|S|=k$. A recent result of Dong and Wu [SIAM J. Discrete Math., 36 (2022) 229--240] shows that every…