Euclid Early Release Observations - The formation of peanutty dwarf galaxies in Perseus

Context. Dwarf galaxies in dense cluster environments are susceptible to tidal interactions that can alter their morphology and kinematics. Boxy isophotes are well studied in massive galaxies but remain poorly understood in dwarfs. Aims. We aim to identify and characterise boxy/peanutty dwarf galaxies in the Perseus cluster and determine the origin of their isophotal shapes. Methods. Using Euclid Early Release Observations of the Perseus cluster, we present a cumulative light fraction method for robustly measuring the isophotal shape parameter c4, particularly suited to low surface brightness regimes. From the similar to 1100 catalogued dwarfs, we select a clean sample of similar to 190 early-type systems sufficiently resolved and free of contamination for reliable c4 measurement. Observed trends are interpreted through comparison with mock Euclid observations of gravitational N-body simulations of tidally transformed dwarfs. Results. From this clean sample, we identify 13 dwarfs with significantly boxy isophotes (c4 < -0.0175), confirmed through visual inspection. These galaxies lack visible thin disks, lie on the cluster red sequence, and show no preferential spatial concentration within Perseus (Kolmogorov-Smirnov p = 0.74 versus the full dE sample). We find a significant anticorrelation between c4 and effective radius: larger galaxies exhibit more boxy (more negative c4) isophotes. We caution that the linear (Pearson) coefficient is sensitive to the two most boxy galaxies; the rank-based (Spearman) coefficient remains significant (rs = -0.59, p = 0.035). The parameter c4 shows no correlation with S & eacute;rsic index or concentration. An analogous size-shape anticorrelation is recovered in the simulations, where inner regions are dominated by box orbits associated with a triaxial peanut structure and outer regions by short-axis tube orbits. Conclusions. Boxy/peanutty dwarf galaxies in Perseus are tidally transformed remnants of moderately rotating progenitors, with boxy isophotes tracing inner box-orbit-dominated peanut structures. The size-shape correlation arises from viewing geometry: face-on orientations reveal the rectangular profile of the elongated triaxial structure (large and boxy), while edge-on views yield rounder, compact morphologies. Our sample thus represents an orientation-selected subset of tidally transformed, peanutty dwarfs in the cluster.