In this paper, we introduce new types of approximate palindromes called single-arm-gapped palindromes (shortly SAGPs). A SAGP contains a gap in either its left or right arm, which is in the form of either $wguc u^R w^R$ or $wuc u^Rgw^R$, where $w$ and $u$ are non-empty strings, $w^R$ and $u^R$ are respectively the reversed strings of $w$ and $u$, $g$ is a string called a gap, and $c$ is either a single character or the empty string. Here we call $wu$ and $u^R w^R$ the arm of the SAGP, and $|uv|$ the length of the arm. We classify SAGPs into two groups: those which have $ucu^R$ as a maximal palindrome (type-1), and the others (type-2). We propose several algorithms to compute type-1 SAGPs with longest arms occurring in a given string, based on suffix arrays. Then, we propose a linear-time algorithm to compute all type-1 SAGPs with longest arms, based on suffix trees. Also, we show how to compute type-2 SAGPs with longest arms in linear time. We also perform some preliminary experiments to show practical performances of the proposed methods.