From 7af7d27d4d4a5491641eac7e844d76b2220a93ec Mon Sep 17 00:00:00 2001 From: William Kenyon <115789274+William-Kenyon@users.noreply.github.com> Date: Thu, 24 Sep 2026 20:40:52 +0100 Subject: [PATCH] Fixes issue #942, adding checks for if a tournament is paradoxical and k-paradoxical --- doc/oper.xml | 28 ++++++++++++++++++++++++++++ doc/prop.xml | 28 ++++++++++++++++++++++++++++ doc/z-chap4.xml | 1 + doc/z-chap5.xml | 1 + gap/oper.gd | 1 + gap/oper.gi | 38 ++++++++++++++++++++++++++++++++++++++ gap/prop.gd | 1 + gap/prop.gi | 14 ++++++++++++++ tst/standard/oper.tst | 24 ++++++++++++++++++++++++ tst/standard/prop.tst | 18 ++++++++++++++++++ 10 files changed, 154 insertions(+) diff --git a/doc/oper.xml b/doc/oper.xml index daa733164..b0c016c19 100644 --- a/doc/oper.xml +++ b/doc/oper.xml @@ -2133,6 +2133,34 @@ true <#/GAPDoc> +<#GAPDoc Label="IsKParadoxical"> + + + true or false. + + A tournament D is k-paradoxical if for every set + S of k vertices there is a vertex outside S which + is an in-neighbour of every vertex in S, i.e. there is one player + who beats all the players of S. This is a generalisation of + . +

+ If D has no vertices, then the answer is false. If D + has at least one vertex and k is 0 then the answer is + true. +

+ If k is negative or D is not a tournament (see + ), then an error is raised. + D := Digraph([[2, 3, 5], [3, 4, 6], [4, 5, 7], [5, 6, 1], [6, 7, 2], +> [7, 1, 3], [1, 2, 4]]);; +gap> IsKParadoxical(D, 2); +true +gap> IsKParadoxical(D, 3); +false]]> + + +<#/GAPDoc> + <#GAPDoc Label="DigraphDijkstra"> diff --git a/doc/prop.xml b/doc/prop.xml index 8a7b39790..f4107a4bb 100644 --- a/doc/prop.xml +++ b/doc/prop.xml @@ -451,6 +451,34 @@ false <#/GAPDoc> +<#GAPDoc Label="IsParadoxical"> + + + true or false. + + A tournament D is paradoxical if every vertex has at least + one in-neighbour, i.e. each player loses at least one game. This is the + same as being 1-paradoxical; see . +

+ If D has no vertices, then the answer is false. +

+ If D is not a tournament (see ), then an + error is raised. +

+ + &MUTABLE_RECOMPUTED_PROP; + + D := Digraph([[2], [3], [1]]);; +gap> IsParadoxical(D); +true +gap> D := Digraph([[2, 3], [3], []]);; +gap> IsParadoxical(D); +false]]> + + +<#/GAPDoc> + <#GAPDoc Label="IsChainDigraph"> diff --git a/doc/z-chap4.xml b/doc/z-chap4.xml index 059e10eea..0b12cb846 100644 --- a/doc/z-chap4.xml +++ b/doc/z-chap4.xml @@ -18,6 +18,7 @@ <#Include Label="DigraphOutEdges"> <#Include Label="IsDigraphEdge"> <#Include Label="IsMatching"> + <#Include Label="IsKParadoxical"> <#Include Label="DigraphMaximalMatching"> <#Include Label="DigraphMaximumMatching"> diff --git a/doc/z-chap5.xml b/doc/z-chap5.xml index 491deb4c8..820c795bf 100644 --- a/doc/z-chap5.xml +++ b/doc/z-chap5.xml @@ -21,6 +21,7 @@ <#Include Label="IsReflexiveDigraph"> <#Include Label="IsSymmetricDigraph"> <#Include Label="IsTournament"> + <#Include Label="IsParadoxical"> <#Include Label="IsTransitiveDigraph"> diff --git a/gap/oper.gd b/gap/oper.gd index 4766b3496..0a47c5bf2 100644 --- a/gap/oper.gd +++ b/gap/oper.gd @@ -117,6 +117,7 @@ DeclareOperation("IsPerfectMatching", [IsDigraph, IsHomogeneousList]); DeclareOperation("IsDigraphPath", [IsDigraph, IsHomogeneousList, IsHomogeneousList]); DeclareOperation("IsDigraphPath", [IsDigraph, IsList]); +DeclareOperation("IsKParadoxical", [IsDigraph, IsInt]); # 9. Connectivity . . . DeclareOperation("DigraphIsKing", [IsDigraph, IsPosInt, IsPosInt]); diff --git a/gap/oper.gi b/gap/oper.gi index bdc462bee..93b378963 100644 --- a/gap/oper.gi +++ b/gap/oper.gi @@ -1596,6 +1596,44 @@ function(D, edges) return true; end); +InstallMethod(IsKParadoxical, "for a tournament and a non-negative int", +[IsDigraph, IsInt], +function(D, k) + local n, subset, blists; + n := DigraphNrVertices(D); + + if not IsTournament(D) then + ErrorNoReturn("the 1st argument must be a tournament,"); + fi; + + if k < 0 then + ErrorNoReturn("the 2nd argument must be non-negative,"); + fi; + + if n = 0 then + return false; + fi; + + if k = 0 then + return true; + fi; + + # this minimum number of vertices needed for k-paradox is given by + # Szekeres, E.; Szekeres, G. (1965), "On a problem of Schütte and Erdős" + if n < (k+2) * (2^(k-1)) - 1 then + return false; + fi; + + blists := List(InNeighbours(D), inn -> BlistList([1..n], inn)); + for subset in Combinations(blists, k) do + if SizeBlist(IntersectionBlist(subset)) = 0 then + return false; + fi; + od; + + return true; +end); + ############################################################################# # 9. Connectivity ############################################################################# diff --git a/gap/prop.gd b/gap/prop.gd index 808d67dc7..bbf2c134e 100644 --- a/gap/prop.gd +++ b/gap/prop.gd @@ -26,6 +26,7 @@ DeclareProperty("IsCompleteBipartiteDigraph", IsDigraph); DeclareProperty("IsCompleteMultipartiteDigraph", IsDigraph); DeclareProperty("IsCompleteDigraph", IsDigraph); DeclareProperty("IsTournament", IsDigraph); +DeclareProperty("IsParadoxical", IsDigraph); DeclareProperty("IsChainDigraph", IsDigraph); DeclareProperty("IsCycleDigraph", IsDigraph); DeclareProperty("IsDigraphCore", IsDigraph); diff --git a/gap/prop.gi b/gap/prop.gi index 0c564ef17..e37e31e5d 100644 --- a/gap/prop.gi +++ b/gap/prop.gi @@ -339,6 +339,20 @@ function(D) return IsAntisymmetricDigraph(D); end); +InstallMethod(IsParadoxical, "for a tournament", +[IsDigraph], +function(D) + if not IsTournament(D) then + ErrorNoReturn("the argument must be a tournament,"); + fi; + + if DigraphNrVertices(D) = 0 then + return false; + else + return not ForAny(InNeighbours(D), IsEmpty); + fi; +end); + InstallMethod(IsEmptyDigraph, "for a digraph with known number of edges", [IsDigraph and HasDigraphNrEdges], D -> DigraphNrEdges(D) = 0); diff --git a/tst/standard/oper.tst b/tst/standard/oper.tst index 585f10a99..ef2c82ee4 100644 --- a/tst/standard/oper.tst +++ b/tst/standard/oper.tst @@ -41,6 +41,30 @@ gap> gr := Digraph([[], [3, 4], [1, 4], [1]]); gap> DigraphIsKing(gr, 2, 2); Error, the 1st argument must be a tournament, +# IsKParadoxical: for a tournament and a non-negative integer +gap> gr := Digraph([[2], [1]]);; +gap> IsKParadoxical(gr, 1); +Error, the 1st argument must be a tournament, +gap> gr := EmptyDigraph(0);; +gap> IsKParadoxical(gr, 1); +false +gap> gr := Digraph([[]]);; +gap> IsKParadoxical(gr, 0); +true +gap> IsKParadoxical(gr, -1); +Error, the 2nd argument must be non-negative, +gap> gr := Digraph([[2], [3], [1]]);; +gap> IsTournament(gr); +true +gap> IsKParadoxical(gr, 1); +true +gap> IsKParadoxical(gr, 2); +false +gap> gr := Digraph([[2, 3, 5], [3, 4, 6], [4, 5, 7], [5, 6, 1], [6, 7, 2], +> [7, 1, 3], [1, 2, 4]]);; +gap> IsKParadoxical(gr, 2); +true + # DigraphRemoveLoops gap> gr := DigraphFromDigraph6String("&EhxPC?@"); diff --git a/tst/standard/prop.tst b/tst/standard/prop.tst index 623ff6713..7055af9f4 100644 --- a/tst/standard/prop.tst +++ b/tst/standard/prop.tst @@ -416,6 +416,24 @@ gap> gr := Digraph([[2], [1], [1]]); gap> IsTournament(gr); false +# IsParadoxical +gap> gr := Digraph([[2], [1]]);; +gap> IsParadoxical(gr); +Error, the argument must be a tournament, +gap> gr := EmptyDigraph(0);; +gap> IsParadoxical(gr); +false +gap> gr := Digraph([[2, 3], [3], []]);; +gap> IsTournament(gr); +true +gap> IsParadoxical(gr); +false +gap> gr := Digraph([[2], [3], [1]]);; +gap> IsTournament(gr); +true +gap> IsParadoxical(gr); +true + # IsStronglyConnectedDigraph gap> gr := Digraph([]);