## 1 Introduction  
Recently, hyperbolic metamaterials (HMMs) have attracted great attention due to their open iso-frequency contour (IFC) in the wave vector space. Once the topological transition from a closed ellipsoid to an open hyperboloid has been completed, the interaction between light and matter is significant changed, for example through enhanced spontaneous emission [1; 2; 3; 4], all-angle negative refraction [5; 6; 7; 8; 9], super-resolution imaging [10; 11; 12; 13; 14], long-range interaction [15; 16; 17], nonlinear super-resolution imaging [18], and Cherenkov emissions with low energy electrons [19], etc. To date, by varying the frequency, the sign of the real part of permittivity (\(\epsilon\)) or permeability (\(\mu\)) in dispersive metamaterials is changed and thereby the topological transition of dispersion is realized [1; 8; 20; 21]. In addition, by maintaining the real parts and tuning the imaginary part of \(\varepsilon\) or \(\mu\) at a fixed frequency, the topological transition of the IFC has been theoretically analyzed [22] and experimentally demonstrated in metamaterials with added losses [23; 24]. In contrast to the passive metamaterials, the study of actively tuned metamaterials and meta-devices is also an outstanding research topic [25; 26; 27; 28]. Actively controlled metamaterial systems have been predicted to be able to yield new applications ranging from electrically switchable devices [27] to tunable coupling devices [26; 28]. The realization of the actively tunable topological transition of the IFC is also very useful in the design of new active optical devices. To realize the externally tunable behaviors, two-dimensional (2D) semiconductors such as graphene are usually utilized. Graphene, a 2D honeycomb lattice of carbon atoms, is electrically characterized by its surface conductivity \(\sigma(\omega,\mu_{c})\), where \(\mu_{c}\) is the chemical potential that can be tuned by the gate voltage [25], and \(\omega\) denotes the angular frequency. Tunable absorption [29; 30], giant Purcell effect [31], tunable invisibility cloaking of particles by means of graphene coatings [32; 33], and hyperbolic plasmonics [34; 35] have been theoretically proposed in graphene-based structures associated with the hyperbolic property in suitable conditions. In this paper, by tuning the chemical potential of the graphene layer, we theoretically analyze and numerically simulate the tunable emission patterns in a periodic structure composed of graphene/dielectric multilayers. The emission patterns clearly show the topological transition of the IFC from a closed ellipsoid to an open hyperboloid [20; 24]. Remarkably, by actively changing the chemical potential of the graphene layer, two special cases corresponding to two kinds of anisotropic \(\varepsilon\)-near-zero (ENZ) media can be realized in the proposed structure at a fixed frequency. To observe the influence of the topological transition on the wave propagations, we study the transmission property of the two kinds of ENZ media embedded with a perfect electric conductor (PEC) or perfect magnetic conductor (PMC) defect. Furthermore, we propose a microwave experimental platform to demonstrate the actively controlled topological transition of the IFC. By adding electrically controlled diodes into a two-dimensional (2D) transmission line (TL) system, we experimentally observe the actively tuned topological transition from the emission patterns by modulating the external voltages applied in the diodes. The paper is organized as follows. In Section 2, by tuning the chemical potential of graphene layer, we show an actively adjustable topological transition of dispersion in the periodic structure composed of graphene and dielectric layers. Moreover, the influences of the topological transition of dispersion on the scattering properties of structures with embedded PEC or PMC defects have been fully studied.