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Euler number of a binary image is a fundamental topological feature that remains invariant under translation, rotation, scaling, and rubber-sheet transformation of the image. In this work, a run-based method for computing Euler number is formulated and a new hardware implementation is described. Analysis of time complexity and performance measure is provided to demonstrate the efficiency of the method. The sequential version of the proposed algorithm requires significantly fewer number of pixel accesses compared to the existing methods and tools based on bit-quad counting or quad-tree, both for the worst case and the average case. A pipelined architecture is designed with a single adder tree to implement the algorithm on-chip by exploiting its inherent parallelism. The architecture uses O(N) 2-input gates and requires O(N log N) time to compute the Euler number of an N · N image. The same hardware, with minor modification, can be used to handle arbitrarily large pixel matrices. A standard cell based VLSI implementation of the architecture is also reported. As Euler number is a widely used parameter, the proposed design can be readily used to save computation time in many image processing applications.

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a b s t r a c tA subset D ⊆ V of a graph G = (V , E) is a (1, j)-set (Chellali et al., 2013) if every vertex v ∈ V \D is adjacent to at least 1 but not more than j vertices in D. The cardinality of a minimum (1, j)-set of G, denoted as γ (1,j) (G), is called the (1, j)-domination number of G. In this paper, using probabilistic methods, we obtain an upper bound on γ (1,j) (G) for j ≥ O(log ∆), where ∆ is the maximum degree of the graph. The proof of this upper bound yields a randomized linear time algorithm. We show that the associated decision problem is NP-complete for choral graphs but, answering a question of Chellali et al., provide a linear-time algorithm for trees for a fixed j. Apart from this, we design a polynomial time algorithm for finding γ (1,j) (G) for a fixed j in a split graph, and show that (1, j)-set problem is fixed parameter tractable in bounded genus graphs and bounded treewidth graphs.

This paper develops a 3GPP-inspired design for the 1 in-band-full-duplex (IBFD) integrated access and backhaul (IAB) 2 networks in the frequency range 2 (FR2) band, which can 3 enhance the spectral efficiency (SE) and coverage while reducing 4 the latency. However, the self-interference (SI), which is usually 5 more than 100 dB higher than the signal-of-interest, becomes the 6 major bottleneck in developing these IBFD networks. We design 7 and analyze a subarray-based hybrid beamforming IBFD-IAB 8 system with the RF beamformers obtained via RF codebooks 9 given by a modified Linde-Buzo-Gray (LBG) algorithm. The 10 SI is canceled in three stages, where the first stage of antenna 11 isolation is assumed to be successfully deployed. The second stage 12 consists of the optical domain (OD)-based RF cancellation, where 13 cancelers are connected with the RF chain pairs. The third stage 14 is comprised of the digital cancellation via successive interference 15 cancellation followed by minimum mean-squared error baseband 16 receiver. Multiuser interference in the access link is canceled by 17 zero-forcing at the IAB-node transmitter. Simulations show that 18 under 400 MHz bandwidth, our proposed OD-based RF cancel-19 lation can achieve around 25 dB of cancellation with 100 taps. 20 Moreover, the higher the hardware impairment and channel 21 estimation error, the worse digital cancellation ability we can 22 obtain. 23 Index Terms-Wideband in-band-full-duplex millimeter wave 24 (FR2 band), subarray hybrid beamforming, integrated access and 25 backhaul, codebook design, self-interference cancellation. 26 I. INTRODUCTION 27 F REQUENCY range 2 (FR2) band (i.e., millimeter wave) 28 communications have been identified as the key technol-29 ogy for the beyond fifth-generation (5G) wireless communi-30 cations to provide much larger bandwidth, narrower beam, 31