The weakly connected independent set polytope in corona and join of graphs

Given a connected undirected graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V, E)$$\end{document}, a subset W of nodes of G is a weakly connected independent set if W is an independent set and the partial graph (V,δ(W))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(V, \delta (W))$$\end{document} is connected, where δ(W)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta (W)$$\end{document} is the set of edges with only one endnode in W. This article proposes several distinct results about the weakly connected independent sets of a graph obtained by corona or join operations: the complete description of the wcis polytope for a corona or a join of two graphs of which we know the wcis polytopes or the maximal independent set polytopes, and the consequences of these graph operations on the minimum weight weakly connected independent set problem (MWWCISP). A class of graphs is also inductively defined from the connected bipartite graphs, the cycles and the strongly chordal graphs, for which the MWWCISP is polynomially solvable. This work is a direct continuation of the article (Bendali et al. in Discrete Optim 22:87–110, 2016) where a similar theorem about complete description of the wcis polytope has been given for 1-sum operation.