/* Part of SWI-Prolog Author: R.A.O'Keefe, Vitor Santos Costa, Jan Wielemaker E-mail: J.Wielemaker@vu.nl WWW: http://www.swi-prolog.org Copyright (c) 1984-2023, VU University Amsterdam CWI, Amsterdam SWI-Prolog Solutions .b.v All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */ :- module(ugraphs, [ add_edges/3, % +Graph, +Edges, -NewGraph add_vertices/3, % +Graph, +Vertices, -NewGraph complement/2, % +Graph, -NewGraph compose/3, % +LeftGraph, +RightGraph, -NewGraph del_edges/3, % +Graph, +Edges, -NewGraph del_vertices/3, % +Graph, +Vertices, -NewGraph edges/2, % +Graph, -Edges neighbors/3, % +Vertex, +Graph, -Vertices neighbours/3, % +Vertex, +Graph, -Vertices reachable/3, % +Vertex, +Graph, -Vertices top_sort/2, % +Graph, -Sort ugraph_layers/2, % +Graph, -Layers transitive_closure/2, % +Graph, -Closure transpose_ugraph/2, % +Graph, -NewGraph vertices/2, % +Graph, -Vertices vertices_edges_to_ugraph/3, % +Vertices, +Edges, -Graph ugraph_union/3, % +Graph1, +Graph2, -Graph connect_ugraph/3 % +Graph1, -Start, -Graph ]). /** Graph manipulation library The S-representation of a graph is a list of (vertex-neighbours) pairs, where the pairs are in standard order (as produced by keysort) and the neighbours of each vertex are also in standard order (as produced by sort). This form is convenient for many calculations. A new UGraph from raw data can be created using vertices_edges_to_ugraph/3. Adapted to support some of the functionality of the SICStus ugraphs library by Vitor Santos Costa. Ported from YAP 5.0.1 to SWI-Prolog by Jan Wielemaker. @author R.A.O'Keefe @author Vitor Santos Costa @author Jan Wielemaker @license BSD-2 or Artistic 2.0 */ :- autoload(library(lists),[append/3]). :- autoload(library(ordsets), [ord_subtract/3,ord_union/3,ord_add_element/3,ord_union/4]). :- autoload(library(error), [instantiation_error/1]). %! vertices(+Graph, -Vertices) % % Unify Vertices with all vertices appearing in Graph. Example: % % ?- vertices([1-[3,5],2-[4],3-[],4-[5],5-[]], L). % L = [1, 2, 3, 4, 5] vertices([], []) :- !. vertices([Vertex-_|Graph], [Vertex|Vertices]) :- vertices(Graph, Vertices). %! vertices_edges_to_ugraph(+Vertices, +Edges, -UGraph) is det. % % Create a UGraph from Vertices and edges. Given a graph with a % set of Vertices and a set of Edges, Graph must unify with the % corresponding S-representation. Note that the vertices without % edges will appear in Vertices but not in Edges. Moreover, it is % sufficient for a vertice to appear in Edges. % % == % ?- vertices_edges_to_ugraph([],[1-3,2-4,4-5,1-5], L). % L = [1-[3,5], 2-[4], 3-[], 4-[5], 5-[]] % == % % In this case all vertices are defined implicitly. The next % example shows three unconnected vertices: % % == % ?- vertices_edges_to_ugraph([6,7,8],[1-3,2-4,4-5,1-5], L). % L = [1-[3,5], 2-[4], 3-[], 4-[5], 5-[], 6-[], 7-[], 8-[]] % == vertices_edges_to_ugraph(Vertices, Edges, Graph) :- sort(Edges, EdgeSet), p_to_s_vertices(EdgeSet, IVertexBag), append(Vertices, IVertexBag, VertexBag), sort(VertexBag, VertexSet), p_to_s_group(VertexSet, EdgeSet, Graph). %! add_vertices(+Graph, +Vertices, -NewGraph) % % Unify NewGraph with a new graph obtained by adding the list of % Vertices to Graph. Example: % % ``` % ?- add_vertices([1-[3,5],2-[]], [0,1,2,9], NG). % NG = [0-[], 1-[3,5], 2-[], 9-[]] % ``` add_vertices(Graph, Vertices, NewGraph) :- msort(Vertices, V1), add_vertices_to_s_graph(V1, Graph, NewGraph). add_vertices_to_s_graph(L, [], NL) :- !, add_empty_vertices(L, NL). add_vertices_to_s_graph([], L, L) :- !. add_vertices_to_s_graph([V1|VL], [V-Edges|G], NGL) :- compare(Res, V1, V), add_vertices_to_s_graph(Res, V1, VL, V, Edges, G, NGL). add_vertices_to_s_graph(=, _, VL, V, Edges, G, [V-Edges|NGL]) :- add_vertices_to_s_graph(VL, G, NGL). add_vertices_to_s_graph(<, V1, VL, V, Edges, G, [V1-[]|NGL]) :- add_vertices_to_s_graph(VL, [V-Edges|G], NGL). add_vertices_to_s_graph(>, V1, VL, V, Edges, G, [V-Edges|NGL]) :- add_vertices_to_s_graph([V1|VL], G, NGL). add_empty_vertices([], []). add_empty_vertices([V|G], [V-[]|NG]) :- add_empty_vertices(G, NG). %! del_vertices(+Graph, +Vertices, -NewGraph) is det. % % Unify NewGraph with a new graph obtained by deleting the list of % Vertices and all the edges that start from or go to a vertex in % Vertices to the Graph. Example: % % == % ?- del_vertices([1-[3,5],2-[4],3-[],4-[5],5-[],6-[],7-[2,6],8-[]], % [2,1], % NL). % NL = [3-[],4-[5],5-[],6-[],7-[6],8-[]] % == % % @compat Upto 5.6.48 the argument order was (+Vertices, +Graph, % -NewGraph). Both YAP and SWI-Prolog have changed the argument % order for compatibility with recent SICStus as well as % consistency with del_edges/3. del_vertices(Graph, Vertices, NewGraph) :- sort(Vertices, V1), % JW: was msort ( V1 = [] -> Graph = NewGraph ; del_vertices(Graph, V1, V1, NewGraph) ). del_vertices(G, [], V1, NG) :- !, del_remaining_edges_for_vertices(G, V1, NG). del_vertices([], _, _, []). del_vertices([V-Edges|G], [V0|Vs], V1, NG) :- compare(Res, V, V0), split_on_del_vertices(Res, V,Edges, [V0|Vs], NVs, V1, NG, NGr), del_vertices(G, NVs, V1, NGr). del_remaining_edges_for_vertices([], _, []). del_remaining_edges_for_vertices([V0-Edges|G], V1, [V0-NEdges|NG]) :- ord_subtract(Edges, V1, NEdges), del_remaining_edges_for_vertices(G, V1, NG). split_on_del_vertices(<, V, Edges, Vs, Vs, V1, [V-NEdges|NG], NG) :- ord_subtract(Edges, V1, NEdges). split_on_del_vertices(>, V, Edges, [_|Vs], Vs, V1, [V-NEdges|NG], NG) :- ord_subtract(Edges, V1, NEdges). split_on_del_vertices(=, _, _, [_|Vs], Vs, _, NG, NG). %! add_edges(+Graph, +Edges, -NewGraph) % % Unify NewGraph with a new graph obtained by adding the list of Edges % to Graph. Example: % % ``` % ?- add_edges([1-[3,5],2-[4],3-[],4-[5], % 5-[],6-[],7-[],8-[]], % [1-6,2-3,3-2,5-7,3-2,4-5], % NL). % NL = [1-[3,5,6], 2-[3,4], 3-[2], 4-[5], % 5-[7], 6-[], 7-[], 8-[]] % ``` add_edges(Graph, Edges, NewGraph) :- p_to_s_graph(Edges, G1), ugraph_union(Graph, G1, NewGraph). %! ugraph_union(+Graph1, +Graph2, -NewGraph) % % NewGraph is the union of Graph1 and Graph2. Example: % % ``` % ?- ugraph_union([1-[2],2-[3]],[2-[4],3-[1,2,4]],L). % L = [1-[2], 2-[3,4], 3-[1,2,4]] % ``` ugraph_union(Set1, [], Set1) :- !. ugraph_union([], Set2, Set2) :- !. ugraph_union([Head1-E1|Tail1], [Head2-E2|Tail2], Union) :- compare(Order, Head1, Head2), ugraph_union(Order, Head1-E1, Tail1, Head2-E2, Tail2, Union). ugraph_union(=, Head-E1, Tail1, _-E2, Tail2, [Head-Es|Union]) :- ord_union(E1, E2, Es), ugraph_union(Tail1, Tail2, Union). ugraph_union(<, Head1, Tail1, Head2, Tail2, [Head1|Union]) :- ugraph_union(Tail1, [Head2|Tail2], Union). ugraph_union(>, Head1, Tail1, Head2, Tail2, [Head2|Union]) :- ugraph_union([Head1|Tail1], Tail2, Union). %! del_edges(+Graph, +Edges, -NewGraph) % % Unify NewGraph with a new graph obtained by removing the list of % Edges from Graph. Notice that no vertices are deleted. Example: % % ``` % ?- del_edges([1-[3,5],2-[4],3-[],4-[5],5-[],6-[],7-[],8-[]], % [1-6,2-3,3-2,5-7,3-2,4-5,1-3], % NL). % NL = [1-[5],2-[4],3-[],4-[],5-[],6-[],7-[],8-[]] % ``` del_edges(Graph, Edges, NewGraph) :- p_to_s_graph(Edges, G1), graph_subtract(Graph, G1, NewGraph). %! graph_subtract(+Set1, +Set2, ?Difference) % % Is based on ord_subtract graph_subtract(Set1, [], Set1) :- !. graph_subtract([], _, []). graph_subtract([Head1-E1|Tail1], [Head2-E2|Tail2], Difference) :- compare(Order, Head1, Head2), graph_subtract(Order, Head1-E1, Tail1, Head2-E2, Tail2, Difference). graph_subtract(=, H-E1, Tail1, _-E2, Tail2, [H-E|Difference]) :- ord_subtract(E1,E2,E), graph_subtract(Tail1, Tail2, Difference). graph_subtract(<, Head1, Tail1, Head2, Tail2, [Head1|Difference]) :- graph_subtract(Tail1, [Head2|Tail2], Difference). graph_subtract(>, Head1, Tail1, _, Tail2, Difference) :- graph_subtract([Head1|Tail1], Tail2, Difference). %! edges(+Graph, -Edges) % % Unify Edges with all edges appearing in Graph. Example: % % ?- edges([1-[3,5],2-[4],3-[],4-[5],5-[]], L). % L = [1-3, 1-5, 2-4, 4-5] edges(Graph, Edges) :- s_to_p_graph(Graph, Edges). p_to_s_graph(P_Graph, S_Graph) :- sort(P_Graph, EdgeSet), p_to_s_vertices(EdgeSet, VertexBag), sort(VertexBag, VertexSet), p_to_s_group(VertexSet, EdgeSet, S_Graph). p_to_s_vertices([], []). p_to_s_vertices([A-Z|Edges], [A,Z|Vertices]) :- p_to_s_vertices(Edges, Vertices). p_to_s_group([], _, []). p_to_s_group([Vertex|Vertices], EdgeSet, [Vertex-Neibs|G]) :- p_to_s_group(EdgeSet, Vertex, Neibs, RestEdges), p_to_s_group(Vertices, RestEdges, G). p_to_s_group([V1-X|Edges], V2, [X|Neibs], RestEdges) :- V1 == V2, !, p_to_s_group(Edges, V2, Neibs, RestEdges). p_to_s_group(Edges, _, [], Edges). s_to_p_graph([], []) :- !. s_to_p_graph([Vertex-Neibs|G], P_Graph) :- s_to_p_graph(Neibs, Vertex, P_Graph, Rest_P_Graph), s_to_p_graph(G, Rest_P_Graph). s_to_p_graph([], _, P_Graph, P_Graph) :- !. s_to_p_graph([Neib|Neibs], Vertex, [Vertex-Neib|P], Rest_P) :- s_to_p_graph(Neibs, Vertex, P, Rest_P). %! transitive_closure(+Graph, -Closure) % % Generate the graph Closure as the transitive closure of Graph. % Example: % % ``` % ?- transitive_closure([1-[2,3],2-[4,5],4-[6]],L). % L = [1-[2,3,4,5,6], 2-[4,5,6], 4-[6]] % ``` transitive_closure(Graph, Closure) :- warshall(Graph, Graph, Closure). warshall([], Closure, Closure) :- !. warshall([V-_|G], E, Closure) :- memberchk(V-Y, E), % Y := E(v) warshall(E, V, Y, NewE), warshall(G, NewE, Closure). warshall([X-Neibs|G], V, Y, [X-NewNeibs|NewG]) :- memberchk(V, Neibs), !, ord_union(Neibs, Y, NewNeibs), warshall(G, V, Y, NewG). warshall([X-Neibs|G], V, Y, [X-Neibs|NewG]) :- !, warshall(G, V, Y, NewG). warshall([], _, _, []). %! transpose_ugraph(Graph, NewGraph) is det. % % Unify NewGraph with a new graph obtained from Graph by replacing % all edges of the form V1-V2 by edges of the form V2-V1. The cost % is O(|V|*log(|V|)). Notice that an undirected graph is its own % transpose. Example: % % == % ?- transpose([1-[3,5],2-[4],3-[],4-[5], % 5-[],6-[],7-[],8-[]], NL). % NL = [1-[],2-[],3-[1],4-[2],5-[1,4],6-[],7-[],8-[]] % == % % @compat This predicate used to be known as transpose/2. % Following SICStus 4, we reserve transpose/2 for matrix % transposition and renamed ugraph transposition to % transpose_ugraph/2. transpose_ugraph(Graph, NewGraph) :- edges(Graph, Edges), vertices(Graph, Vertices), flip_edges(Edges, TransposedEdges), vertices_edges_to_ugraph(Vertices, TransposedEdges, NewGraph). flip_edges([], []). flip_edges([Key-Val|Pairs], [Val-Key|Flipped]) :- flip_edges(Pairs, Flipped). %! compose(+LeftGraph, +RightGraph, -NewGraph) % % Compose NewGraph by connecting the _drains_ of LeftGraph to the % _sources_ of RightGraph. Example: % % ?- compose([1-[2],2-[3]],[2-[4],3-[1,2,4]],L). % L = [1-[4], 2-[1,2,4], 3-[]] compose(G1, G2, Composition) :- vertices(G1, V1), vertices(G2, V2), ord_union(V1, V2, V), compose(V, G1, G2, Composition). compose([], _, _, []) :- !. compose([Vertex|Vertices], [Vertex-Neibs|G1], G2, [Vertex-Comp|Composition]) :- !, compose1(Neibs, G2, [], Comp), compose(Vertices, G1, G2, Composition). compose([Vertex|Vertices], G1, G2, [Vertex-[]|Composition]) :- compose(Vertices, G1, G2, Composition). compose1([V1|Vs1], [V2-N2|G2], SoFar, Comp) :- compare(Rel, V1, V2), !, compose1(Rel, V1, Vs1, V2, N2, G2, SoFar, Comp). compose1(_, _, Comp, Comp). compose1(<, _, Vs1, V2, N2, G2, SoFar, Comp) :- !, compose1(Vs1, [V2-N2|G2], SoFar, Comp). compose1(>, V1, Vs1, _, _, G2, SoFar, Comp) :- !, compose1([V1|Vs1], G2, SoFar, Comp). compose1(=, V1, Vs1, V1, N2, G2, SoFar, Comp) :- ord_union(N2, SoFar, Next), compose1(Vs1, G2, Next, Comp). %! ugraph_layers(Graph, -Layers) is semidet. %! top_sort(+Graph, -Sorted) is semidet. % % Sort vertices topologically. Layers is a list of lists of vertices % where there are no edges from a layer to an earlier layer. The % predicate top_sort/2 flattens the layers using append/2. % % These predicates fail if Graph is cyclic. If Graph is not connected, % the sub-graphs are individually sorted, where the root of each % subgraph is in the first layer, the nodes connected to the roots in % the second, etc. % % ``` % ?- top_sort([1-[2], 2-[3], 3-[]], L). % L = [1, 2, 3] % ``` % % @compat The original version of this library provided top_sort/3 as % a _difference list_ version of top_sort/2. We removed this because % the argument order was non-standard. Fixing causes hard to debug % compatibility issues while we expect top_sort/3 was rarely used. A % backward compatible top_sort/3 can be defined as % % ``` % top_sort(Graph, Tail, Sorted) :- % top_sort(Graph, Sorted0), % append(Sorted0, Tail, Sorted). % ``` % % The original version returned all vertices in a _layer_ in reverse % order. The current one returns them in standard order of terms, % i.e., each layer is an _ordered set_. % % @compat ugraph_layers/2 is a SWI-Prolog specific addition to this % library. top_sort(Graph, Sorted) :- ugraph_layers(Graph, Layers), append(Layers, Sorted). ugraph_layers(Graph, Layers) :- vertices_and_zeros(Graph, Vertices, Counts0), count_edges(Graph, Vertices, Counts0, Counts1), select_zeros(Counts1, Vertices, Zeros), top_sort(Zeros, Layers, Graph, Vertices, Counts1). vertices_and_zeros([], [], []) :- !. vertices_and_zeros([Vertex-_|Graph], [Vertex|Vertices], [0|Zeros]) :- vertices_and_zeros(Graph, Vertices, Zeros). % Count the number of incomming edges for each vertex count_edges([], _, Counts, Counts) :- !. count_edges([_-Neibs|Graph], Vertices, Counts0, Counts2) :- incr_list(Neibs, Vertices, Counts0, Counts1), count_edges(Graph, Vertices, Counts1, Counts2). incr_list([], _, Counts, Counts) :- !. incr_list([V1|Neibs], [V2|Vertices], [M|Counts0], [N|Counts1]) :- V1 == V2, !, N is M+1, incr_list(Neibs, Vertices, Counts0, Counts1). incr_list(Neibs, [_|Vertices], [N|Counts0], [N|Counts1]) :- incr_list(Neibs, Vertices, Counts0, Counts1). % get the vertices with 0 incoming edges, i.e., the origins. select_zeros([], [], []) :- !. select_zeros([0|Counts], [Vertex|Vertices], [Vertex|Zeros]) :- !, select_zeros(Counts, Vertices, Zeros). select_zeros([_|Counts], [_|Vertices], Zeros) :- select_zeros(Counts, Vertices, Zeros). %! top_sort(+Zeros, -Layers, +Graph, +Vertices, +Counts) is semidet. top_sort([], Layers, Graph, _, Counts) :- !, vertices_and_zeros(Graph, _, Counts), % verify nothing left Layers = []. top_sort(Zeros, [Zeros|Layers], Graph, Vertices, Counts1) :- decr_zero_neighbors(Zeros, Graph, Vertices, Counts1, Counts2, NewZeros, []), top_sort(NewZeros, Layers, Graph, Vertices, Counts2). decr_zero_neighbors([], _, _, Counts, Counts, Z, Z). decr_zero_neighbors([Zero|Zeros], Graph, Vertices, Counts0, Counts, Z0, Z) :- graph_memberchk(Zero-Neibs, Graph), decr_list(Neibs, Vertices, Counts0, Counts1, Z0, Z1), decr_zero_neighbors(Zeros, Graph, Vertices, Counts1, Counts, Z1, Z). graph_memberchk(Element1-Edges, [Element2-Edges2|_]) :- Element1 == Element2, !, Edges = Edges2. graph_memberchk(Element, [_|Rest]) :- graph_memberchk(Element, Rest). decr_list([], _, Counts, Counts, Zeros, Zeros) :- !. decr_list([V1|Neibs], [V2|Vertices], [N|Counts1], [M|Counts2], Z0, Z) :- V1 == V2, !, M is N - 1, ( M == 0 -> Z0 = [V1|Z1], decr_list(Neibs, Vertices, Counts1, Counts2, Z1, Z) ; decr_list(Neibs, Vertices, Counts1, Counts2, Z0, Z) ). decr_list(Neibs, [_|Vertices], [N|Counts1], [N|Counts2], Zi, Zo) :- decr_list(Neibs, Vertices, Counts1, Counts2, Zi, Zo). %! neighbors(+Vertex, +Graph, -Neigbours) is det. %! neighbours(+Vertex, +Graph, -Neigbours) is det. % % Neigbours is a sorted list of the neighbours of Vertex in Graph. % Example: % % ``` % ?- neighbours(4,[1-[3,5],2-[4],3-[], % 4-[1,2,7,5],5-[],6-[],7-[],8-[]], NL). % NL = [1,2,7,5] % ``` neighbors(Vertex, Graph, Neig) :- neighbours(Vertex, Graph, Neig). neighbours(V,[V0-Neig|_],Neig) :- V == V0, !. neighbours(V,[_|G],Neig) :- neighbours(V,G,Neig). %! connect_ugraph(+UGraphIn, -Start, -UGraphOut) is det. % % Adds Start as an additional vertex that is connected to all vertices % in UGraphIn. This can be used to create an topological sort for a % not connected graph. Start is before any vertex in UGraphIn in the % standard order of terms. No vertex in UGraphIn can be a variable. % % Can be used to order a not-connected graph as follows: % % ``` % top_sort_unconnected(Graph, Vertices) :- % ( top_sort(Graph, Vertices) % -> true % ; connect_ugraph(Graph, Start, Connected), % top_sort(Connected, Ordered0), % Ordered0 = [Start|Vertices] % ). % ``` connect_ugraph([], 0, []) :- !. connect_ugraph(Graph, Start, [Start-Vertices|Graph]) :- vertices(Graph, Vertices), Vertices = [First|_], before(First, Start). %! before(+Term, -Before) is det. % % Unify Before to a term that comes before Term in the standard % order of terms. % % @error instantiation_error if Term is unbound. before(X, _) :- var(X), !, instantiation_error(X). before(Number, Start) :- number(Number), !, Start is Number - 1. before(_, 0). %! complement(+UGraphIn, -UGraphOut) % % UGraphOut is a ugraph with an edge between all vertices that are % _not_ connected in UGraphIn and all edges from UGraphIn removed. % Example: % % ``` % ?- complement([1-[3,5],2-[4],3-[], % 4-[1,2,7,5],5-[],6-[],7-[],8-[]], NL). % NL = [1-[2,4,6,7,8],2-[1,3,5,6,7,8],3-[1,2,4,5,6,7,8], % 4-[3,5,6,8],5-[1,2,3,4,6,7,8],6-[1,2,3,4,5,7,8], % 7-[1,2,3,4,5,6,8],8-[1,2,3,4,5,6,7]] % ``` % % @tbd Simple two-step algorithm. You could be smarter, I suppose. complement(G, NG) :- vertices(G,Vs), complement(G,Vs,NG). complement([], _, []). complement([V-Ns|G], Vs, [V-INs|NG]) :- ord_add_element(Ns,V,Ns1), ord_subtract(Vs,Ns1,INs), complement(G, Vs, NG). %! reachable(+Vertex, +UGraph, -Vertices) % % True when Vertices is an ordered set of vertices reachable in % UGraph, including Vertex. Example: % % ?- reachable(1,[1-[3,5],2-[4],3-[],4-[5],5-[]],V). % V = [1, 3, 5] reachable(N, G, Rs) :- reachable([N], G, [N], Rs). reachable([], _, Rs, Rs). reachable([N|Ns], G, Rs0, RsF) :- neighbours(N, G, Nei), ord_union(Rs0, Nei, Rs1, D), append(Ns, D, Nsi), reachable(Nsi, G, Rs1, RsF).