Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs

Constructive linear time algorithms for branchwidth

“…Using Lemma 14 we can easily verify that Des R (B) is a branch model of G rooted on R and we call it description of B with respect to R. We set C R (B) = C(Des R (B)) and we call C R (B) characteristic of B with respect to R. Clearly, C R (B) is dense and typical and is an ancestor of Des R (B). Very similarly to [6] and [7] one can prove the following useful lemmata. …”

“…As an example we mention that if A = (5,5,6,7,7,7,4,4,3,5,4,6,8,2,9,3,4,6,7,2,7,5,4,4,6,4), then τ (A) = (5, 7, 3, 8, 2, 9, 2, 7, 4). We call a sequence A typical if τ (A) = A i.e.…”

“…We also say that A ≺< B if there exist a C ∈ S such that A ≺ C and C < B. The proof of the following Lemma can be found in [6] (Lemma 3.19).…”

Imbalance Is Fixed Parameter Tractable

“…This is achieved in 2 It is well-known that every problem that is FPT admits a kernel (see [15] for definition). While graph layout problems such as Treewidth, Pathwidth and Cutwidth are FPT [3,5,17] one can easily show using recently developed machinery [4] that they are unlikely to admit polynomial kernels. Giving such a lower bound for Imbalance seems non-trivial, while a polynomial kernel for the problem would be the first such kernel for a graph layout problem.…”

Integer Maximum Flow in Wireless Sensor Networks with Energy Constraint

“…A small modification of the construction in [6] shows that we can obtain in linear time, given a tree decomposition of width at most k of a graph G, a nice tree decomposition of G of width at most k, such that the root node r has X r = {s}.…”