On Noncrossing and Plane Tree-Like Structures

Year 2021, Volume: 4 Issue: 2, 89 - 99, 30.06.2021

Isaac Owino Okoth

https://doi.org/10.33434/cams.803065

https://izlik.org/JA22WS94JK

Abstract

Mathematical trees are connected graphs without cycles, loops and multiple edges. Various trees such as Cayley trees, plane trees, binary trees, $d$-ary trees, noncrossing trees among others have been studied extensively. Tree-like structures such as Husimi graphs and cacti are graphs which posses the conditions for trees if, instead of vertices, we consider their blocks. In this paper, we use generating functions and bijections to find formulas for the number of noncrossing Husimi graphs, noncrossing cacti and noncrossing oriented cacti. We extend the work to obtain formulas for the number of bicoloured noncrossing Husimi graphs, bicoloured noncrossing cacti and bicoloured noncrossing oriented cacti. Finally, we enumerate plane Husimi graphs, plane cacti and plane oriented cacti according to number of blocks, block types and leaves.

Keywords

noncrossing tree , plane trees , tree-like structure , Husimi graph , cactus , enumeration

Supporting Institution

Maseno University

Thanks

We thenk DergiPark for allowing us to publish in your journal.

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