Light graphs in families of outerplanar graphs

We prove that every 2-connected outerplanar graph of order at least k   (k⩾3k⩾3) contains a path on k   vertices with all vertices of degree at most k+3k+3 and a path on k   vertices with degree sum at most 4k-24k-2. Further, every 2-connected outerplanar graph without adjacent vertices with degree sum ⩽5⩽5 contains a triangle C3C3 with vertices of degrees 2, 4 and b  , 4⩽b⩽64⩽b⩽6, and a 3-star K1,3K1,3 with central vertex of degree 4 and remaining vertices of degrees 2, 2 and b  , 4⩽b⩽64⩽b⩽6. Moreover, all bounds are best possible.