A vertex subset D of G is a dominating set if every vertex in V(G)∖D is adjacent to a vertex in D. A dominating set D is independent if G[D], the subgraph of G induced by D, contains no edge. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set of G, and the independent domination number i(G) of G is the minimum cardinality of an independent dominating set of G. A classical work related to the relationship between γ(G) and i(G) of a graph G was established in 1978 by Allan and Laskar. They proved that every K1,3-free graph G satisfies γ(G)=i(H). Hexagonal systems (2 connected planar graphs whose interior faces are all hexagons) have been extensively studied as they are used to present bezenoid hydrocarbon structures which play an important role in organic chemistry. The domination numbers of hexagonal systems have been studied continuously since 2018 when Hutchinson et al. posted conjectures, generated from a computer program called Conjecturing, related to the domination numbers of hexagonal systems. Very recently in 2021, Bermudo et al. answered all of these conjectures. In this paper, we extend these studies by considering the relationship between the domination number and the independent domination number of hexagonal systems. Although every hexagonal system H with at least two hexagons contains K1,3 as an induced subgraph, we find many classes of hexagonal systems whose domination number is equal to an independent domination number. However, we establish the existence of a hexagonal system H such that γ(H)<i(H) with the prescribed number of hexagons.

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A graph G is said to be k-γc-critical if the connected domination number γc(G) is equal to k and γc(G+uv)<k for any pair of non-adjacent vertices u and v of G. Let ζ be the number of cut vertices of G and let ζ0 be the maximum number of cut vertices that can be contained in one block. For an integer ℓ≥0, a graph G is ℓ-factor critical if G−S has a perfect matching for any subset S of vertices of size ℓ. It was proved by Ananchuen in 2007 for k=3, Kaemawichanurat and Ananchuen in 2010 for k=4 and by Kaemawichanurat and Ananchuen in 2020 for k≥5 that every k-γc-critical graph has at most k−2 cut vertices and the graphs with maximum number of cut vertices were characterized. In 2020, Kaemawichanurat and Ananchuen proved further that, for k≥4, every k-γc-critical graphs satisfies the inequality ζ0(G)≤mink+23,ζ. In this paper, we characterize all k-γc-critical graphs having k−3 cut vertices. Further, we establish realizability that, for given k≥4, 2≤ζ≤k−2 and 2≤ζ0≤mink+23,ζ, there exists a k-γc-critical graph with ζ cut vertices having a block which contains ζ0 cut vertices. Finally, we proved that every k-γc-critical graph of odd order with minimum degree two is 1-factor critical if and only if 1≤k≤2. Further, we proved that every k-γc-critical K1,3-free graph of even order with minimum degree three is 2-factor critical if and only if 1≤k≤2.

Background: Glanzmann\&apos;s thrombasthenia (GT) is a rare congenital bleeding disorder clinically presented with mucocutaneous bleeding associated with trauma and/or surgery. Patients with GT have normal platelet count but prolonged bleeding time. GT is been reported to be associated with mutations in the genes, which encode for glycoprotein IIb/IIIa (GPIIb/IIIa). Case presentation: A 2-year-old male patient with a history of recurrent nasal bleeding for 1 year was presented to us. Bleeding time was found prolonged (9 minutes), while activated partial thromboplastin time was 37 seconds, prothrombin time (PT) was 13.5 seconds and remained within the normal range. Platelet aggregation assays were defective when using adenosine diphosphate, adrenaline, collagen, and arachidonic acid. Genetic analysis found a novel likely pathogenic homozygous mutation c.985G &gt; T in the ITGA2B gene. The subjects were controlled by using 1 g of aminocaproic acid twice daily for 10 days, which improved the bleeding time was improved to 6 minutes. Conclusion: The present study reported a child (2 years) with novel pathogenic mutation c.985G &gt; T in the ITGA2B gene associated with GT and reviewed its clinical management.

A k-γ c -edge critical graph is a graph G with the connected domination number γ ccritical graph is a graph which are both k-γ c -edge critical and k-γ c -vertex critical. Let κ, δ , ω and α be respectively connectivity minimum degree, clique number and independence number. In this paper, we prove that every maximal 3-γ c -vertex critical graph G satisfies α ≤ δ and this bound is best possible. We prove further that G satisfies α + ω ≤ n − 1 and we also characterize all such graphs achieving the upper bounds. We finally show that if G satisfies κ < δ , then every two vertices of G are joined by hamiltonian path.