TY - JOUR

T1 - Listing minimal edge-covers of intersecting families with applications to connectivity problems

AU - Nutov, Zeev

PY - 2009/1/6

Y1 - 2009/1/6

N2 - Let G = (V, E) be a directed/undirected graph, let s, t ∈ V, and let F be an intersecting family on V (that is, X ∩ Y, X ∪ Y ∈ F for any intersecting X, Y ∈ F) so that s ∈ X and t ∉ X for every X ∈ F. An edge set I ⊆ E is an edge-cover of F if for every X ∈ F there is an edge in I from X to V - X. We show that minimal edge-covers of F can be listed with polynomial delay, provided that, for any I ⊆ E the minimal member of the residual family FI of the sets in F not covered by I can be computed in polynomial time. As an application, we show that minimal undirected Steiner networks, and minimal k-connected and k-outconnected spanning subgraphs of a given directed/undirected graph, can be listed in incremental polynomial time.

AB - Let G = (V, E) be a directed/undirected graph, let s, t ∈ V, and let F be an intersecting family on V (that is, X ∩ Y, X ∪ Y ∈ F for any intersecting X, Y ∈ F) so that s ∈ X and t ∉ X for every X ∈ F. An edge set I ⊆ E is an edge-cover of F if for every X ∈ F there is an edge in I from X to V - X. We show that minimal edge-covers of F can be listed with polynomial delay, provided that, for any I ⊆ E the minimal member of the residual family FI of the sets in F not covered by I can be computed in polynomial time. As an application, we show that minimal undirected Steiner networks, and minimal k-connected and k-outconnected spanning subgraphs of a given directed/undirected graph, can be listed in incremental polynomial time.

KW - Intersecting families

KW - Listing

KW - Minimal edge-covers

KW - Steiner network

UR - http://www.scopus.com/inward/record.url?scp=56349094673&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2008.04.026

DO - 10.1016/j.dam.2008.04.026

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AN - SCOPUS:56349094673

SN - 0166-218X

VL - 157

SP - 112

EP - 117

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

IS - 1

ER -