Extremal Edge General Position Sets in Some Graphs

Abstract

A set of edges X⊆E(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X\\subseteq E(G)$$\\end{document} of a graph G is an edge general position set if no three edges from X lie on a common shortest path. The edge general position number gpe(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{gp}}_{\ extrm{e}}(G)$$\\end{document} of G is the cardinality of a largest edge general position set in G. Graphs G with gpe(G)=|E(G)|-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{gp}}_{{\ extrm{e}}}(G) = |E(G)| - 1$$\\end{document} and with gpe(G)=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{gp}}_{{\ extrm{e}}}(G) = 3$$\\end{document} are respectively characterized. Sharp upper and lower bounds on gpe(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{gp}}_{{\ extrm{e}}}(G)$$\\end{document} are proved for block graphs G and exact values are determined for several specific block graphs.