Algorithms for hard problems Parse trees for graphs Juris Viksna, 2015

t-boundaried graphs [Adapted from R.Downey, M.Fellows]

t-boundaried graphs [Adapted from R.Downey, M.Fellows]

Gluing of t-boundaried graphs [Adapted from R.Downey, M.Fellows]

Gluing of t-boundaried graphs [Adapted from R.Downey, M.Fellows] Note about the notation: x refers to the 1st graph x  refers to the 2nd graph x refers to the resulting graph

Graph parsing operators for tw=2 [Adapted from R.Downey, M.Fellows]

Graph parsing operators for tw=2 [Adapted from R.Downey, M.Fellows] Comment It seems that gluing operator is additionally needed – otherwise not clear how to obtain even K 3. The tree also does not produce G – probably some b-s should actually be gluings and maybe there are some mistakes in argument order.

Graph parsing operators for tw=2 [Adapted from R.Downey, M.Fellows]

Graph parsing operators [Adapted from R.Downey, M.Fellows]

Parsing theorem [Adapted from R.Downey, M.Fellows] Basically the theorem tells us that there is a one-to-(many?) correspondence between graphs of treewidth t and trees produced by parsing operators of size t+1. More over, graph  tree correspondence is constructive and could be computed in linear time. Still, sometimes we might need to be a bit careful, since tree  graph correspondence is given up to graph isomorphism.

Parsing theorem [Adapted from R.Downey,M.Fellows]

Parsing theorem [Adapted from R.Downey,M.Fellows]

Parsing theorem [Adapted from R.Downey, M.Fellows]

Parsing theorem [Adapted from R.Downey, M.Fellows]

Parsing theorem [Adapted from R.Downey, M.Fellows]

Parsing theorem [Adapted from R.Downey, M.Fellows]

Parsing theorem [Adapted from R.Downey, M.Fellows]

Parsing theorem [Adapted from R.Downey, M.Fellows]

Parsing theorem [Adapted from R.Downey,M.Fellows]

Parsing theorem [Adapted from R.Downey,M.Fellows]

Parsing theorem [Adapted from R.Downey, M.Fellows]

Parsing theorem [Adapted from R.Downey, M.Fellows]

“Small” and “Large” universes [Adapted from R.Downey, M.Fellows] Generally we will now be interested in properties of particular classes of t-boundaried graphs of treewidth t (we include the parse operator in definition to get boundary that can be “obtained” via parsing process). Sometimes this parsability may not matter for graphs of interest and we can consider simply the large universe of t-boundaried garphs.

Parsing replacement property [Adapted from R.Downey, M.Fellows]

Parsing replacement property [Adapted from R.Downey, M.Fellows]

Parsing replacement property [Adapted from R.Downey, M.Fellows]

Congruences and equivalences for graphs [Adapted from R.Downey, M.Fellows]

Finite state collections of parse graphs [Adapted from R.Downey, M.Fellows] So, we are already quite close to the generalization of FPT algorithms based on FA on strings :) Collection F is t-finite state, if parse trees for all graphs from F can be recognized by FTA. Thus, membership of F can be solved by construction of FTA (time f(k)) and running it on a given instance of graph (time O(n)).

Myhill-Nerode theorem for parse graphs [Adapted from R.Downey, M.Fellows] Thus, the main point is translation from graph congruences with finite index to finite state trees.

Myhill-Nerode theorem for parse graphs [Adapted from R.Downey, M.Fellows] Generally we will now be interested in properties of particular classes of t-boundaried graphs of treewidth t (we include the parse operator in definition to get boundary that can be “obtained” via parsing process). Sometimes this parsability may not matter for graphs of interest and we can consider simply the large universe of t-boundaried garphs.

Myhill-Nerode theorem for parse graphs [Adapted from R.Downey, M.Fellows]

Myhill-Nerode theorem for parse graphs [Adapted from R.Downey, M.Fellows]

Myhill-Nerode theorem for parse graphs [Adapted from R.Downey, M.Fellows]

Myhill-Nerode theorem for parse graphs [Adapted from R.Downey, M.Fellows]

Myhill-Nerode theorem for parse graphs [Adapted from R.Downey, M.Fellows]

Myhill-Nerode theorem for parse graphs [Adapted from R.Downey, M.Fellows]

Myhill-Nerode theorem for parse graphs [Adapted from R.Downey, M.Fellows]

Myhill-Nerode theorem for parse graphs [Adapted from R.Downey, M.Fellows]

Graphs with bounded bandwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded bandwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded bandwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded bandwidth [Adapted from R.Downey, M.Fellows]

Test sets and Hamiltonicity [Adapted from R.Downey, M.Fellows]

Test sets and Hamiltonicity [Adapted from R.Downey, M.Fellows]

Test sets and Hamiltonicity [Adapted from R.Downey, M.Fellows]

Test sets and Hamiltonicity [Adapted from R.Downey, M.Fellows]

Graphs with bounded cutwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded cutwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded cutwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded cutwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded cutwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded cutwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded cutwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded cutwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded cutwidth [Adapted from R.Downey, M.Fellows]

Graphs with bounded cutwidth [Adapted from R.Downey, M.Fellows]