Hamiltonian chordal graphs are not cycle extendible

In 1990, Hendry conjectured that every Hamiltonian chordal graph is cycle extendible; that is, the vertices of any non-Hamiltonian cycle are contained in a cycle of length one greater. We disprove this conjecture by constructing counterexamples on $n$ vertices for any $n \geq 15$. Furthermore, we show that there exist counterexamples where the ratio of the length of a non-extendible cycle to the total number of vertices can be made arbitrarily small. We then consider cycle extendibility in Hamiltonian chordal graphs where certain induced subgraphs are forbidden, notably $P_n$ and the bull.Comment: Some results from Section 3 were incorrect and have been removed. To appear in SIAM Journal on Discrete Mathematic

Pi/2-Angle Yao Graphs are Spanners

We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor 8(29+23sqrt(2)). Enroute to this, we also show that the Yao graph Y4 in the Linf metric is a planar spanner with stretch factor 8.Comment: 20 pages, 9 figure

A method for eternally dominating strong grids

International audienceIn the eternal domination game, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices and no more than one guard may occupy a vertex. The goal is to determine the eternal domination number of a graph which is the minimum number of guards required to defend the graph against an infinite sequence of attacks. In this paper, we continue the study of the eternal domination game on strong grids. Cartesian grids have been vastly studied with tight bounds for small grids such as 2×n, 3×n, 4×n, and 5×n grids, and recently it was proven in [Lamprou et al., CIAC 2017, 393-404] that the eternal domination number of these grids in general is within O(m + n) of their domination number which lower bounds the eternal domination number. Recently, Finbow et al. proved that the eternal domination number of strong grids is upper bounded by mn 6 + O(m + n). We adapt the techniques of [Lamprou et al., CIAC 2017, 393-404] to prove that the eternal domination number of strong grids is upper bounded by mn 7 + O(m + n). While this does not improve upon a recently announced bound of ⎡m/3⎤ x⎡n/3⎤ + O(m √ n) [Mc Inerney, Nisse, Pérennes, HAL archives, 2018; Mc Inerney, Nisse, Pérennes, CIAC 2019] in the general case, we show that our bound is an improvement in the case where the smaller of the two dimensions is at most 6179

A method for eternally dominating strong grids

International audienceIn the eternal domination game, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices and no more than one guard may occupy a vertex. The goal is to determine the eternal domination number of a graph which is the minimum number of guards required to defend the graph against an infinite sequence of attacks. In this paper, we continue the study of the eternal domination game on strong grids. Cartesian grids have been vastly studied with tight bounds for small grids such as 2×n, 3×n, 4×n, and 5×n grids, and recently it was proven in [Lamprou et al., CIAC 2017, 393-404] that the eternal domination number of these grids in general is within O(m + n) of their domination number which lower bounds the eternal domination number. Recently, Finbow et al. proved that the eternal domination number of strong grids is upper bounded by mn 6 + O(m + n). We adapt the techniques of [Lamprou et al., CIAC 2017, 393-404] to prove that the eternal domination number of strong grids is upper bounded by mn 7 + O(m + n). While this does not improve upon a recently announced bound of ⎡m/3⎤ x⎡n/3⎤ + O(m √ n) [Mc Inerney, Nisse, Pérennes, HAL archives, 2018; Mc Inerney, Nisse, Pérennes, CIAC 2019] in the general case, we show that our bound is an improvement in the case where the smaller of the two dimensions is at most 6179