Minimum broadcast tree decompositions

Minimum Size Tree-decompositions

We study in this paper the problem of computing a tree-decomposition of a graph with width at most k and minimum number of bags. More precisely, we focus on the following problem: given a fixed k ≥ 1, what is the complexity of computing a treedecomposition of width at most k with minimum number of bags in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for...

Minimum tree decompositions with a given tree as a factor

Girth, minimum degree, independence, and broadcast independence

An independent broadcast on a connected graph $G$is a function $f:V(G)to mathbb{N}_0$such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$,and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$.The broadcast independence number $alpha_b(G)$ of $G$is the largest weight $sumlimits_{xin V(G)}f(x)$of an ind...

Simplicial Decompositions, Tree-decompositions and Graph Minors

The concepts of simplicial decompositions, tree-decompositions and simplicial tree-decompositions were all inspired by a common forerunner: the decompositions of finite graphs used by K. Wagner in his classic paper [ 13 ], in which he proved the equivalence of the 4-Colour-Conjecture to Hadwiger’s Conjecture for n = 5. To show that the 4CC implies Hadwiger’s Conjecture (for n = 5), Wagner used ...

Reconfiguration over Tree Decompositions

A vertex-subset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. The reconfiguration version of a vertex-subset problem Q asks whether it is possible to transform one feasible solution for Q into another in at most l steps, where each step is a vertex addition or deletion, and each intermediate set is also a feasible solution for Q of size bounded ...