Table 2 Examples of gadgets \(\mathcal {G}_e=(e\cup {\hat{\mathcal {V}}}, \mathcal {E}')\) incorporating EDVWs

(EDVWs-based) Clique gadget

Undirected

\(\emptyset\)

\(|e|(|e|-1)/2\)

For every \(u,v\in e\), add an edge (u, v) of weight \(\gamma _e(u)\gamma _e(v)\).

(EDVWs-based) Star gadget

Undirected

\(\{v_e\}\)

|e|

For every \(v\in e\), add an edge \((v,v_e)\) of weight \(\gamma _e(v)\).

Sym. EDVWs-based gadget

Directed

\(\{e',e''\}\)

\(2|e|+1\)

For every \(v\in e\), add a directed edge of weight \(\gamma _e(v)\) from v to \(e'\) and a directed edge of weight \(\gamma _e(v)\) from \(e''\) to v. Moreover, add a directed edge of weight b from \(e'\) to \(e''\).

Asym. EDVWs-based gadget

Directed

\(\{v_e\}\)

2|e|

For every \(v\in e\), add a directed edge of weight \(a\gamma _e(v)\) from v to \(v_e\) and a directed edge of weight \(b\gamma _e(v)\) from \(v_e\) to v.