Does this graph construction have a name?











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For a given graph $G = (V, E)$, construct a graph $G' = (V', E')$ where





  • $V' = V uplus V$ (disjoint union)


  • $( (b, v), (d, u) ) in E'$ (with $b,din mathbb B$) iff $(v,u)in E$ and either $b = d$ or $vneq u$


Essentially, it is the union of the graph products $G,Box, (0;;1)$ and $Gtimes (0{-}1)$. Two copies of $G$ with every pair of vertices connected according to $G$'s layout, excepting a direct connection between any pair of vertices.



I'm looking to use it to define boolean satisfiability probles in terms of tolerance relations/dependency graphs, so each vertex of $G$ represents a variable and each vertex of $G'$ represents assigning a variable to 0 or 1.



ETA: $mathbb B = {0, 1}$, the boolean domain. Used in the definition of disjoint union: $Auplus B = Atimes {0} cup Btimes {1} subseteq (Acup B)times mathbb B$.






graph-theory







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  • What is $mathbb B$?
    – Servaes
    yesterday












  • @Servaes $mathbb B = {0,1}$.
    – Karl Damgaard Asmussen
    yesterday

















up vote
0
down vote

favorite












For a given graph $G = (V, E)$, construct a graph $G' = (V', E')$ where





  • $V' = V uplus V$ (disjoint union)


  • $( (b, v), (d, u) ) in E'$ (with $b,din mathbb B$) iff $(v,u)in E$ and either $b = d$ or $vneq u$


Essentially, it is the union of the graph products $G,Box, (0;;1)$ and $Gtimes (0{-}1)$. Two copies of $G$ with every pair of vertices connected according to $G$'s layout, excepting a direct connection between any pair of vertices.



I'm looking to use it to define boolean satisfiability probles in terms of tolerance relations/dependency graphs, so each vertex of $G$ represents a variable and each vertex of $G'$ represents assigning a variable to 0 or 1.



ETA: $mathbb B = {0, 1}$, the boolean domain. Used in the definition of disjoint union: $Auplus B = Atimes {0} cup Btimes {1} subseteq (Acup B)times mathbb B$.






graph-theory







share|cite|improve this question
























  • What is $mathbb B$?
    – Servaes
    yesterday












  • @Servaes $mathbb B = {0,1}$.
    – Karl Damgaard Asmussen
    yesterday















up vote
0
down vote

favorite









up vote
0
down vote

favorite











For a given graph $G = (V, E)$, construct a graph $G' = (V', E')$ where





  • $V' = V uplus V$ (disjoint union)


  • $( (b, v), (d, u) ) in E'$ (with $b,din mathbb B$) iff $(v,u)in E$ and either $b = d$ or $vneq u$


Essentially, it is the union of the graph products $G,Box, (0;;1)$ and $Gtimes (0{-}1)$. Two copies of $G$ with every pair of vertices connected according to $G$'s layout, excepting a direct connection between any pair of vertices.



I'm looking to use it to define boolean satisfiability probles in terms of tolerance relations/dependency graphs, so each vertex of $G$ represents a variable and each vertex of $G'$ represents assigning a variable to 0 or 1.



ETA: $mathbb B = {0, 1}$, the boolean domain. Used in the definition of disjoint union: $Auplus B = Atimes {0} cup Btimes {1} subseteq (Acup B)times mathbb B$.






graph-theory







share|cite|improve this question















For a given graph $G = (V, E)$, construct a graph $G' = (V', E')$ where





  • $V' = V uplus V$ (disjoint union)


  • $( (b, v), (d, u) ) in E'$ (with $b,din mathbb B$) iff $(v,u)in E$ and either $b = d$ or $vneq u$


Essentially, it is the union of the graph products $G,Box, (0;;1)$ and $Gtimes (0{-}1)$. Two copies of $G$ with every pair of vertices connected according to $G$'s layout, excepting a direct connection between any pair of vertices.



I'm looking to use it to define boolean satisfiability probles in terms of tolerance relations/dependency graphs, so each vertex of $G$ represents a variable and each vertex of $G'$ represents assigning a variable to 0 or 1.



ETA: $mathbb B = {0, 1}$, the boolean domain. Used in the definition of disjoint union: $Auplus B = Atimes {0} cup Btimes {1} subseteq (Acup B)times mathbb B$.






graph-theory




graph-theory






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edited yesterday

























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Karl Damgaard Asmussen

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  • What is $mathbb B$?
    – Servaes
    yesterday












  • @Servaes $mathbb B = {0,1}$.
    – Karl Damgaard Asmussen
    yesterday




















  • What is $mathbb B$?
    – Servaes
    yesterday












  • @Servaes $mathbb B = {0,1}$.
    – Karl Damgaard Asmussen
    yesterday


















What is $mathbb B$?
– Servaes
yesterday






What is $mathbb B$?
– Servaes
yesterday














@Servaes $mathbb B = {0,1}$.
– Karl Damgaard Asmussen
yesterday






@Servaes $mathbb B = {0,1}$.
– Karl Damgaard Asmussen
yesterday

















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