A　bisection　of　a　graph　is　a　bipartition　of　its　vertex　set　in　which　the　two　classes　differ　in　size　by　at　most　one.　For　a　random　bisection　of　a　graph　with　(Formula　presented.)　edges,　one　expects　(Formula　presented.)　edges　spans　in　one　vertex　class.　Bollobás　and　Scott　asked　for　conditions　that　guarantee　a　bisection　in　which　both　classes　span　at　most　(Formula　presented.)　edges　simultaneously.　Let　(Formula　presented.)　be　integers　with　(Formula　presented.),　and　let　(Formula　presented.)　be　a　graph　with　minimum　degree　(Formula　presented.)　and　(Formula　presented.)　edges.　In　this　article,　we　prove　that　if　(Formula　presented.)　contains　neither　triangle　nor　(Formula　presented.)　as　a　subgraph,　then　(Formula　presented.)　admits　a　bisection　in　which　both　classes　span　at　most　(Formula　presented.)　edges.　We　also　consider　Max-Bisections　of　graphs　without　(Formula　presented.)　and　improve　a　recent　result　of　Hou　and　Yan.　©　2021　Wiley　Periodicals　LLC