From 75a8b3630cbcea3835fd36cb4f05b6003474a1e4 Mon Sep 17 00:00:00 2001
From: Yue Ren <yue.ren.kl@gmail.com>
Date: Thu, 21 Nov 2024 14:37:20 +0000
Subject: [PATCH] new: maximal_edge_pairing

---
 .../Curves/ProjectiveCurve.jl                 | 39 +++++++++++++++++++
 1 file changed, 39 insertions(+)

diff --git a/src/AlgebraicGeometry/Curves/ProjectiveCurve.jl b/src/AlgebraicGeometry/Curves/ProjectiveCurve.jl
index 6afd813ed20..00517df7a60 100644
--- a/src/AlgebraicGeometry/Curves/ProjectiveCurve.jl
+++ b/src/AlgebraicGeometry/Curves/ProjectiveCurve.jl
@@ -152,3 +152,42 @@ function graph_curve_with_vertex_and_edge_ideals(G::Graph)
     edgeIdeals = [ vertexIdeals[src(e)] + vertexIdeals[dst(e)] for e in edges(G)]
     return C, vertexIdeals, radical.(edgeIdeals)  # is radical really necessary?
 end
+
+
+function edge_pairing_candidates(G::Graph, e::Edge)
+    v1, v2 = src(e), dst(e)
+    adj_v1 = filter(v -> v != v2, neighbors(G,v1))
+    adj_v2 = filter(v -> v != v1, neighbors(G,v2))
+    pairing1 = ( (Edge(v1,adj_v1[1]),Edge(v2,adj_v2[1])), (Edge(v1,adj_v1[2]),Edge(v2,adj_v2[2])) )
+    pairing2 = ( (Edge(v1,adj_v1[1]),Edge(v2,adj_v2[2])), (Edge(v1,adj_v1[1]),Edge(v2,adj_v2[2])) )
+    return [pairing1,pairing2]
+end
+
+function maximal_edge_pairing(G::Graph)
+    pGfaces = faces(IncidenceMatrix, tutte_lifting(G), 2)
+    EP = Dict{Edge, Tuple{Tuple{Edge, Edge},Tuple{Edge, Edge}}}()
+    for edge in edges(G)
+        pairing1, pairing2 =  edge_pairing_candidates(G,edge)
+        # take the first pair of edges of each pairing candidate
+        # the second pair of edges is not required
+        # as the first pair is correct if and only if the second pair is
+        edge11 = pairing1[1][1]
+        edge12 = pairing1[1][2]
+        edge21 = pairing2[1][1]
+        edge22 = pairing2[1][2]
+        # for each pair of edges, construct the set of four vertices
+        pairingVertices1 = [src(edge11),dst(edge11),src(edge12),dst(edge12)]
+        pairingVertices2 = [src(edge21),dst(edge21),src(edge22),dst(edge22)]
+        # and check whether there is a tutte lifting facet containing all of them
+        for i in 1:nrows(pGfaces)
+            if all(pGfaces[i,pairingVertices1])
+                EP[edge] = pairing1
+                break
+            elseif all(pGfaces[i,pairingVertices2])
+                EP[edge] = pairing2
+                break
+            end
+        end
+    end
+    return EP
+end