The distinguishing index D (G) of a graph G is the least natural number d such that G has an edge colouring with d colours that is only preserved by the identity automorphism. In this paper we investigate the distinguishing index of the Cartesian product of connected finite graphs. We prove that for every k ≥ 2, the k-th Cartesian power of a connected graph G has distinguishing index equal 2, with the only exception D (K 2 2) = 3. We also prove that if G and H are connected graphs that satisfy the relation 2 ≤ |G| ≤ |H| ≤ 2 |G| 2 G − 1 − |G| + 2, then D (G2H) ≤ 2 unless G2H = K 2 2 .
A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The paired domination number, $$\gamma _{\mathrm{pr}}(G)$$ γ pr ( G ) , of G is the minimum cardinality of a paired dominating set of G. In this paper, we show that if T is a tree of order at least 2, then $$\gamma _{\mathrm{pr}}(T) \le 2\alpha (T) - \varphi (T)$$ γ pr ( T ) ≤ 2 α ( T ) - φ ( T ) where $$\alpha (T)$$ α ( T ) is the independence number and $$\varphi (T)$$ φ ( T ) is the $$P_3$$ P 3 -packing number. We present a tight upper bound on the paired domination number of a tree T in terms of its maximum degree $$\varDelta$$ Δ . For $$\varDelta \ge 1$$ Δ ≥ 1 , we show that if T is a tree of order n with maximum degree $$\varDelta$$ Δ , then $$\gamma _{\mathrm{pr}}(T) \le \left( \frac{5\varDelta -4}{8\varDelta -4} \right) n + \frac{1}{2}n_1(T) + \frac{1}{4}n_2(T) - \left( \frac{\varDelta -2}{4\varDelta -2} \right)$$ γ pr ( T ) ≤ 5 Δ - 4 8 Δ - 4 n + 1 2 n 1 ( T ) + 1 4 n 2 ( T ) - Δ - 2 4 Δ - 2 , where $$n_1(T)$$ n 1 ( T ) and $$n_2(T)$$ n 2 ( T ) denote the number of vertices of degree 1 and 2, respectively, in T. Further, we show that this bound is tight for all $$\varDelta \ge 3$$ Δ ≥ 3 . As a consequence of this result, if T is a tree of order $$n \ge 2$$ n ≥ 2 , then $$\gamma _{\mathrm{pr}}(T) \le \frac{5}{8} n + \frac{1}{2}n_1(T) + \frac{1}{4}n_2(T)$$ γ pr ( T ) ≤ 5 8 n + 1 2 n 1 ( T ) + 1 4 n 2 ( T ) , and this bound is asymptotically best possible.
Introduction. Temporomandibular joint (TMJ) disorders are a common diagnostic problem. No universal radiological parameter of the analysis was introduced. Aim. Comparison of values of selected radiological parameters between asymptomatic patients and those with the TMJ arthropathy. Material and methods. Retrospective analysis of CT scans of patients of the Department of Dental and Maxillofacial Radiology and the Department of Cranio-Maxillofacial Surgery, Oral Surgery and Implantology, Medical University of Warsaw. Patients were divided into two groups: 1. without TMJ disorders, 2. with TMJ dysfunction symptoms. Following parameters of heads of mandible were analyzed bilaterally: shape, anteroposterior and lateromedial dimensions, the distance between lateral points of both heads (HL-HR), distance between a head and the mandibular fossa. The angle between the horizontal axis of the head of mandible and the line drawn by posterior points of heads of mandible was measured. Results. The most common type of the head of mandible in group 1 (40 patients; 13 women, 27 men) was convex (14 patients), in group 2 (16 patients; 14 women, 2 men) – plane (8 patients). Significant differences between groups were obtained for: GL-GP (group 1 – 120.35 mm, group 2 – 115.4 mm), dimensions of heads of mandible: lateromedial – 19.7 mm, 18.14 mm, anteroposterior – 8.03 mm, 7.04 mm for group 1 and 2, respectively. Conclusions. Computed tomography allowed for an accurate analysis of the TMJ components. Measurements of structures discussed in this work should be a part of the diagnosis of patients with TMJ dysfunction.
A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The paired domination number, γ pr (G), of G is the minimum cardinality of a paired dominating set of G. A set of vertices whose removal from G produces a graph without isolated vertices is called a non-isolating set. The minimum cardinality of a non-isolating set of vertices whose removal decreases the paired domination number is the γ − pr -stability of G, denoted st − γpr (G). The paired domination stability of G is the minimum cardinality of a non-isolating set of vertices in G whose removal changes the paired domination number. We establish properties of paired domination stability in graphs. We prove that if G is a connected graph with γ pr (G) ≥ 4, then st − γpr (G) ≤ 2∆(G) where ∆(G) is the maximum degree in G, and we characterize the infinite family of trees that achieve equality in this upper bound.
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