Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Articles in Press 2025 05 01 Characterizations of Additively Graceful Signed Paths and Cycles 14941 10.22049/cco.2025.30365.2436 EN Brian D'Souza School of Physical and Applied Sciences, Goa University, Taleigao Plateau, Goa 403206, India Jessica Pereira School of Physical and Applied Sciences, Goa University, Taleigao Plateau, Goa 403206, India Journal Article 2025 02 18 A $(p,m,n)$ signed graph $S$, is a signed graph of order $p$ with $m$ positive edges and $n$ negative edges. In this paper, we first prove a few basic results on vertex labelings of paths. We use these results and a sequence of lemmas to obtain a characterization of additively graceful signed paths. We prove that, apart from exactly 4 exceptions, additively graceful signed paths are characterized by the signed paths containing at most one negative section with $n \leq 2$. We also establish a characterization of additively graceful signed cycles. We prove that a $(p,m,n)$ signed cycle $S$ is additively graceful if and only if one among the following 4 conditions are satisfied, (a) $n=0$ and $ m\equiv 0$ or $3 \pmod 4$,  (b) $n=1$ and $ m\equiv 1$ or $2 \pmod 4$,  (c) $n=2$, $ m\equiv 1$ or $2 \pmod 4$ and $S$ contains a single negative section,  (d) $S$ is the all negative signed cycle on $C_3$.<br />  https://comb-opt.azaruniv.ac.ir/article_14941_80ee96a6fd90fedcca3108dda98d3ac9.pdf