Hogyan befolyásolja a villamosenergia-hálózatról rendelkezésre álló információ a fizikai támadások által okozott sérülékenységről alkotott képet?

Összefoglaló. A villamosenergia-rendszerek fizikai támadásokkal szembeni ellenálló képessége a közelmúltban világszerte történt események ismeretében egyre nagyobb hangsúlyt kap a tématerület kutatásaiban. Az ilyen eseményekre való megfelelő felkészüléshez elengedhetetlen az üzemeltetett infrastruktúrának, elsősorban annak gyengeségeinek pontos ismerete. A cikkben Magyarország villamosenergia-hálózatának adatai alapján készített súlyozatlan és súlyozott gráfokon végzünk vizsgálatokat, hogy megértsük a különböző stratégia mentén kiválasztott célpontok elleni támadások milyen mértékben csökkentik a topológiai hatékonyságot. A cikk célja egyben a magyar hálózat sérülékenységének általános bemutatása is, mely hasznos bemeneti információ lehet a kockázati tervek elkészítésekor. Summary. Tolerance of the power grid against physical intrusions has gained importance in the light of various attacks that have taken place around the world. To adequately prepare for such events, grid operators have to possess a deep understanding of their infrastructure, more specifically, of its weaknesses. A graph representation of the Hungarian power grid was created in a way that the vertices are generators, transformers, and substations and the edges are high-voltage transmission lines. All transmission and sub-transmission elements were considered, including the 132 kV network as well. The network is subjected to various types of single and double element attacks, objects of which are selected according to different aspects. The vulnerability of the network is measured as a relative drop in efficiency when a vertex or an edge is removed from the network. Efficiency is a measure of the network’s performance, assuming that the efficiency for transmitting electricity between vertices i and j is proportional to the reciprocal of their distance. In this paper, simultaneous removals were considered, arranged into two scenarios (single or double element removal) and a total of 5 cases were carried out (single vertex removal, single edge removal, double vertex removal, double edge removal, single vertex and single edge removal). During the examinations, all possible removal combinations were simulated, thus the 5 cases represent 385, 504, 73920, 128271 and 193797 runs, respectively. After all runs were performed, damage values were determined for random and targeted attacks, and attacks causing maximal damage were also identified. In all cases, damage was calculated for unweighted and weighted networks as well, to enable the comparison of those two models. The aims of this paper are threefold: to perform a general assessment on the vulnerability of the Hungarian power grid against random and targeted attacks; to compare the damage caused by different attack strategies; and to highlight the differences between using unweighted and weighted graphs representations. Random removal of a single vertex or a single edge caused 0.3–0.4% drop in efficiency, respectively, which indicates a high tolerance against such attacks. Damage for random double attacks was still only in the range of 0.6–0.8%, which is acceptable. It was shown that if targets are selected by the attacker based on the betweenness rank of the element, damage would be below the maximal possible values. Comparison of the damage measured in the unweighted and the weighted network representations has shown that damage to the weighted network tends to be bigger for vertex attacks, but the contrary is observed for edge attacks. Numerical differences between the two representations do not show any trend that could be generalised, but in the case of the most vulnerable elements significant differences were found in damage measures, which underlines the importance of using weighted models.

On investigations of graphs preserving the Wiener index upon vertex removal

<abstract><p>In this paper, we present solutions of two open problems regarding the Wiener index $ W(G) $ of a graph $ G $. More precisely, we prove that for any $ r \geq 2 $, there exist infinitely many graphs $ G $ such that $ W(G) = W(G - \{v_1, \ldots, v_r\}) $, where $ v_1, \ldots, v_r $ are $ r $ distinct vertices of $ G $. We also prove that for any $ r \geq 1 $ there exist infinitely many graphs $ G $ such that $ W(G) = W(G - \{v_i\}) $, $ 1 \leq i \leq r $, where $ v_1, \ldots, v_r $ are $ r $ distinct vertices of $ G $.</p></abstract>

Split Domination Vertex Critical and Edge Critical Graphs

A dominating set D of a graph G = (V;E) is a split dominating set ifthe induced graph hV 􀀀 Di is disconnected. The split domination number s(G)is the minimum cardinality of a split domination set. A graph G is called vertexsplit domination critical if s(G􀀀v) s(G) for every vertex v 2 G. A graph G iscalled edge split domination critical if s(G + e) s(G) for every edge e in G. Inthis paper, whether for some standard graphs are split domination vertex critical ornot are investigated and then characterized 2- ns-critical and 3- ns-critical graphswith respect to the diameter of a graph G with vertex removal. Further, it is shownthat there is no existence of s-critical graph for edge addition.

Effect of vertex-removal on game total domination numbers

The total domination game is played on a graph [Formula: see text] by two players, named Dominator and Staller. They alternately select vertices of [Formula: see text]; each chosen vertex totally dominates its neighbors. In this game, each chosen vertex must totally dominates at least one new vertex not totally dominated before. The game ends when all vertices in [Formula: see text] are totally dominated. Dominator’s goal is to finish the game as soon as possible, and Staller’s goal is to prolong it as much as possible. The game total domination number is the number of chosen vertices when both players play optimally, denoted by [Formula: see text] when Dominator starts the game and denoted by [Formula: see text] when Staller starts the game. In this paper, we show that for any graph [Formula: see text] and a vertex [Formula: see text], where [Formula: see text] has no isolated vertex, we have [Formula: see text] and [Formula: see text]. Moreover, all such differences can be realized by some connected graphs.