Let \(\Pi _n=[c_1, c_2, \ldots ,c_n,c_1]\) be a positively oriented Jordan polygon with distinct vertices \(c_k\) occurring in order as \(\Pi _n\) is positively traversed, and let \(\Omega \) be the Jordan domain bounded by \(\Pi _n\) . Let f and F be the harmonic extensions into the open unit disc \(\mathbb {D}\) of the step functions on the unit circle \(\mathbb {T}\) defined by \(\begin{aligned}f(e^{it}) = c_k \;(t_{k-1}< t< t_k)\quad \text {and}\quad F(e^{is}) = c_k,\qquad s_{k-1}<s<s_k,\end{aligned}\) where \(\begin{aligned}0\le t_0<t_1<\cdots<t_{n}=t_0+2\pi \quad \text {and}\quad 0\le s_0<s_1<\cdots <s_{n}=s_0+2\pi .\end{aligned}\) Let \(\zeta _k=e^{it_k}\) and \(\eta _k=e^{is_k}\) , where \(0\le k\le n-1\) . Assume there exists \(r, 1\le r\le n-1, \) such that \(\eta _k=\zeta _{k+r}\) , where \(0\le k\le n-1\) and \(\zeta _{n+j}=\zeta _j\) , \(0\le j\le r-1\) . Let \(G=f-F\) and let \(\omega \) be the (second complex) dilatation of G such that \(|\omega |<1\) in \(\mathbb {D}\) . The results of this paper are: (a) If \(\Omega \) is convex, then G is a univalent starlike harmonic mapping.
(b) If \(r=1\) or \( n-1\) , then G is a harmonic mapping that admits the origin exactly once, counting multiplicity.
(c) If \(2\le r\le n-2\) and if an additional condition is fulfilled, then the same conclusion of (b) holds.
(d) G is \(\lceil n/2 -1\rceil \) -valent in \(\mathbb {D}\) ; in particular, G is a harmonic mapping which is univalent close-to-convex if \(n=4\) and is 2-valent if \(n=5\) or 6.
