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Table 1 Examples of cardinality-based and EDVWs-based splitting functions and their corresponding gadgets where \(\mathcal {S}\) is a subset of the hyperedge e and a, b are positive constants

From: Hypergraph cuts with edge-dependent vertex weights

Cardinality-based

EDVWs-based

Corresponding gadget

\(w_e(\mathcal {S})=|\mathcal {S}|\cdot |e\setminus \mathcal {S}|\)

\(w_e(\mathcal {S})=\gamma _e(\mathcal {S})\cdot \gamma _e(e\setminus \mathcal {S})\)

Clique gadget

\(w_e(\mathcal {S})=\min \{|\mathcal {S}|,|e\setminus \mathcal {S}|\}\)

\(w_e(\mathcal {S})=\min \{\gamma _e(\mathcal {S}),\gamma _e(e\setminus \mathcal {S})\}\)

Star gadget

\(w_e(\mathcal {S})=\min \{|\mathcal {S}|,|e\setminus \mathcal {S}|,b\}\)

\(w_e(\mathcal {S})=\min \{\gamma _e(\mathcal {S}),\gamma _e(e\setminus \mathcal {S}),b\}\)

Sym. cardinality/EDVWs-based gadget

\(w_e(\mathcal {S})=\min \{a|\mathcal {S}|,b|e\setminus \mathcal {S}|\}\)

\(w_e(\mathcal {S})=\min \{a\gamma _e(\mathcal {S}),b\gamma _e(e\setminus \mathcal {S})\}\)

Asym. cardinality/EDVWs-based gadget