For n-vertex graphs with treewidth $$k = O(n^{1/2-\epsilon })$$ k = O ( n 1 / 2 - ϵ ) and an arbitrary $$\epsilon >0$$ ϵ > 0 , we present a word-RAM algorithm to compute vertex separators using only O(n) bits of working memory. As an application of our algorithm, we give an O(1)-approximation algorithm for tree decomposition. Our algorithm computes a tree decomposition in $$c^k n (\log \log n) \log ^* n$$ c k n ( log log n ) log ∗ n time using O(n) bits for some constant $$c > 0$$ c > 0 . Together with the result of Banerjee et al. (Proceedings of 21st international conference on computing and combinatorics (COCOON 2015). LNCS, vol 9198, Springer, pp 349–360, 2015. https://doi.org/10.1007/978-3-319-21398-9_28) we are able to compute a solution for all monadic-second-order problems (MSO) with $$O(n + \tau (k) \cdot p (\log _{p} n) \log n)$$ O ( n + τ ( k ) · p ( log p n ) log n ) bits in $$O(\tau (k) \cdot n^{2 + (2/\log p)})$$ O ( τ ( k ) · n 2 + ( 2 / log p ) ) time where k is the treewidth of the given graph, p is some arbitrary parameter with $$2 \le p \le n$$ 2 ≤ p ≤ n and $$\tau $$ τ is some function depending on the MSO formula. We finally use the tree decomposition obtained by our algorithm to solve Vertex Cover, Independent Set, Dominating Set, MaxCut and q-Coloring by using polynomial time and O(n) bits as long as the treewidth of the graph is smaller than $$c' \log n$$ c ′ log n for some problem dependent constant $$0< c' < 1$$ 0 < c ′ < 1 .

Practical applications that use treewidth algorithms have graphs with treewidth k = O( 3 √ n). Given such n-vertex graphs we present a word-RAM algorithm to compute vertex separators using only O(n) bits of working memory. As an application of our algorithm, we show an O(1)approximation algorithm for tree decomposition. Our algorithm computes a tree decomposition in c k n(log * n) log log n time using O(n) bits for some constant c.We finally show that our tree-decomposition algorithm can be used to solve several monadic second-order problems using O(n) bits as long as the treewidth of the graph is smaller than c log n for some constant 0 < c < 1.

We present a novel space-efficient graph coarsening technique for n-vertex separable graphs G, in particular for planar graphs, called cloud partition, which partitions the vertices V (G) into disjoint sets C of size O(log n) such that each C induces a connected subgraph of G. Using this partition P we construct a so-called structure-maintaining minor F of G via specific contractions within the disjoint sets such that F has O(n/ log n) vertices. The combination of (F, P) is referred to as a cloud decomposition.We call a graph G = (V, E) separable if it admits to an O(n c )-separator theorem for some constant c < 1 meaning there exists a separator S ⊂ V that partitions V into {A, S, B} such that no vertices of A and B are adjacent in G and neither A nor B contain more than c n vertices for a fixed constant c < 1. Due to the last property such separators are called balanced. This famously includes planar graphs, which admit an O( √ n)-separator theorem. For planar graphs we show that a cloud decomposition can be constructed in O(n) time and using O(n) bits. Given a cloud decomposition (F, P) constructed for a planar graph G we are able to find a balanced separator of G in O(n/ log n) time. Contrary to related publications, we do not make use of an embedding of the input graph. This allows us to construct the succinct encoding scheme for planar graphs due to Blelloch and Farzan (CPM 2010) in O(n) time and O(n) bits improving both runtime and space by a factor of Θ(log n). As an additional application of our cloud decomposition we show that a tree decomposition for planar graphs of width O(n 1/2+ ) for any > 0 can be constructed in O(n) bits and a time linear in the size of the tree decomposition. A similar result by Izumi and Otachi (ICALP 2020) constructs a tree decomposition of width O(k √ n log n) for graphs of treewidth k ≤ √ n in sublinear space and polynomial time. Finally, we generalize our cloud decomposition from planar graphs to arbitrary separable graphs.