| Summary: | Given a digital image (or digital object) <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, we address some unsolved problems related to the study of fixed point sets of <i>k</i>-continuous self-maps of <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> from the viewpoints of digital curve and digital surface theory. Consider two simple closed <i>k</i>-curves with <inline-formula><math display="inline"><semantics><msub><mi>l</mi><mi>i</mi></msub></semantics></math></inline-formula> elements in <inline-formula><math display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msup></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow><mo>,</mo><msub><mi>l</mi><mn>1</mn></msub><mo>⪈</mo><msub><mi>l</mi><mn>2</mn></msub><mo>≥</mo><mn>4</mn></mrow></semantics></math></inline-formula>. After initially formulating an alignment of fixed point sets of a digital wedge of these curves, we prove that perfectness of it depends on the numbers <inline-formula><math display="inline"><semantics><mrow><msub><mi>l</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mspace width="3.33333pt"></mspace><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, instead of the <i>k</i>-adjacency. Furthermore, given digital <i>k</i>-surfaces, we also study an alignment of fixed point sets of digital <i>k</i>-surfaces and digital wedges of them. Finally, given a digital image which is not perfect, we explore a certain condition that makes it perfect. In this paper, each digital image <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> is assumed to be <i>k</i>-connected and <inline-formula><math display="inline"><semantics><mrow><msup><mi>X</mi><mo>♯</mo></msup><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> unless stated otherwise.
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